metamath.set_mm_core

en · unofficial · sha256:ae5e99f5e3d086e53e4f115e1a4f01c423e14fe4b14c6357a8d3e16483e58892

spimfv.1 $e |- ( x = y -> ( ph -> ps ) ) $.

Referencing norms: source_statement/8f9c3380f215edac90baf511a26737f31511bd85194fa60c5cefa0ebd162443a

en · unofficial · sha256:4e7e4ec22cac4a34ecf10075ab949abefbf592a20706c3ece8274577d86056fc

pm2.61d $p |- ( ph -> ch ) $= ( wn con1d syld pm2.18d ) ACACFBCABCEGDHI $.

Referencing norms: proof_reference/086600ef9d6c68be05dc6b5a0d60ae7ae1b3648eae5fe13f01be5abdf92c0545, proof_reference/5c6f13ab526aec9983b39a00ba9d39e02db022490c51c2be0ab707a32dabe561, proof_reference/71462b0f53b9311cc6779c6a645b09b6b95bc86b31f52cc7b9c1e6321c5030df, proof_reference/ea7c15ff209c8d851dbd35de65cb6fc3f2938362d4570f0af318c6a5c146cf23, source_statement/e3c41a3fe0c3fa7b91badf49a0c5060ce9e00d5f64535c4547f037c9a250ca45, statement_hypothesis/247b9fd7216a2a724907b74e0cb5a5345219b804727637c1fedf93b6277ab751, statement_hypothesis/fb96f1af12db0b1909b328281f9fd679f1646253c5c7f9e15c67ca6da823853d, statement_uses/41afff3f2248b67b2576f12297c26e58021c77128ee77f79b2fadf440f0c62a2, statement_uses/5d8e40701747143f093893628e06f3ebf4e341cf011725c2516a8a6c3ee33fde, statement_uses/b2e1a65a5140195f15bc09542f3bb293257f9730c052f8a4b3dba82772ab0d3f, statement_uses/bb25f1b2036b244cbd592ca0a7de757b40675242babad09022efa2d5fc2e4851, statement_uses/c32d9ac4a24d314c15ff8a64eccf96fc3ebe3f608aa9aec3439171780b071e1b

en · unofficial · sha256:2e297243898d5438c96880cf2ece84191ddd3c879fed452d3bf7316846ff4585

pm2.65iOLD $p |- -. ph $= ( wn con2i con3i pm2.61i ) BAEABDFABCGH $.

Referencing norms: proof_reference/6edd198af5b251d8fb5c3a1799caf5a690bb7f62b364be32f0ddb970dcd47de7, proof_reference/8501662befce82f5d8710ef69c05bd4dafaf0fd71066a0b7f8330190ea04893a, proof_reference/a6ae1ef57eb607d912075bf3d790cc7667020f34c07a1f4c4f0ebef9dfdb4cc4, proof_reference/c3ac920d55662ff2e17d16dd8879a0b21b307f30d84921738dc4dd89e2b07484, source_statement/324d586c3970e44834fce25ae7c960faffa1946fda9e0d3e0358ea0ba6c5a89a, statement_hypothesis/9a58f87dd9fc422540bdddb29549bbdacec6fb9b13d84328b34d823d6bbc7ac4, statement_hypothesis/ea64fbccc8de2fe3e7e6dfcc118f68c10afbd728099b6e4f454b6dca69b0df8c, statement_uses/0f31638af269e0822036bfa6f3a9615d99aca587c1c9965fd17e4f3aca9e1ba5, statement_uses/4489af10d23d721748de71b674a1fa6ee341eee795af6c88ca63f5703272311c

en · unofficial · sha256:a9c60c089e415b692608ee52445c32db44d70021c492dce519663825f4eec153

3bitr2rd $p |- ( ph -> ( ta <-> ps ) ) $= ( bitr4d bitr2d ) ABDEABCDFGIHJ $.

Referencing norms: proof_reference/61da54d958339b130d97ef52560df27d37ad1beefbebe73b5464b4b84f2ce8ba, proof_reference/b96205d7c5ae52d21b1b99844e55744158e86f9c4017611f7131b4bdfd6ee63b, source_statement/4e70ee2a6507d749641cf485ab370857064b7e4395b5d7cea8f76306047424f0, statement_hypothesis/0363f0c1446f17ed5c05cd1c74adfd90d36176d6ba33b8a590baaebf4551885f, statement_hypothesis/a3b58151cc5df19e10c3e8abb551a49f6b5d38773c283ed9eb5fc099eff62646, statement_hypothesis/ee8a41795d6398d338a05a9be2fa5f3b1fc86e30dc76810b8eaf62e57269ac77, statement_uses/27dec97475b42029ae63175c625c9f0a225883f6576849f393b994a5e7c12490, statement_uses/51d9202a5fc243d0e78883d8541897ae57471078b7cffeb023254c213316c0da, statement_uses/5731ec79ec0d8ac0f93891bba74ebc0f49c89b4286839d139615873f35808d8f, statement_uses/a9deb7dd6cadb5afec39f243bbce056f091d88872c349d5f2a5c54f506e01fcd, statement_uses/e9cf64240272ec3df2cab3638a9c88b302e635b568ed0e8008056d762db2ea7d, statement_uses/f2b39336e9dcabe8c2477b862784550b00bf7e214fe046e70312f17b22f048d5, statement_uses/fa02557ee555405387093daae8f4285f9bb15fca40fd8a4314a6c69daf03e41e

en · unofficial · sha256:083808c7b2346bd206cb8a7442b90589cb58c30012b1b3f7e0cbfd0af899b8e6

$( Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.) $)

Referencing norms: statement_comment/87a896521a41744d22a2f5a436ca868004e6b19195077c164231fc4226e7abd1

en · unofficial · sha256:69635cc4f134a40545cedf366ad99d66f559e136c01074bf8df24e01fd735d9e

sylan2d.2 $e |- ( ph -> ( ( th /\ ch ) -> ta ) ) $.

Referencing norms: source_statement/b544394fb3fa140cc904aee10dde7471156c3711508350aa398b8b978f3cc151

en · unofficial · sha256:3a41f405bb1c01fc5a1ab059b1a6aeb78b71e69e4a5bf76b0174ae6a4d27d77b

nfald.1 $e |- F/ y ph $.

Referencing norms: source_statement/eea6603bbc76631c724501e95f936c083bec67d467d9d89dd03551b90764bd87

en · unofficial · sha256:4b124c9f2ed514e53cf533c166d7b0a243a0e70de0e7abfab3c62388f6ed3ada

pm2.61nii.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/d45ad38c282e22eb144df20ef9e199bcb842941c2d5c32719b48513d9c61f888

en · unofficial · sha256:e02c373749ace05f91254d1882a4acca04a0f15767c7f2725037d8174017611b

a1ii.2 $e |- ps $.

Referencing norms: source_statement/ca5730282a10a16dc7781519457d2a339dd2466cb1750ba8743c599e88b0f0ea

en · unofficial · sha256:01374f5ec485f5fa3ca19796ed66e9d7668bfde93545b1948bcc05e69a7bc98a

$( Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)

Referencing norms: statement_comment/43d96e2d706ccca0367359fc7f08691d2c3b79ba3535d9f1870c8e891dfcf0a4

en · unofficial · sha256:268961d0eabe2cf9f066fad15d0771332a68803442eed77b1fff7e3ebca221d6

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Unique existence: the unique existential quantifier =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/984f9fb40a0c921d35d36c713d48bcfbbb66cabf600ac04051a22ece01483e2c, section_title/7c828ea819d5b94e47e9b6a26b3f5788fb9050d1646b734f33f494ad77faf3f0

en · unofficial · sha256:2b0aa2c9060f6d166d812c11afe6bfb87cdb782e79aeba339eb3ed08e7f17bc9

syl2anc $p |- ( ph -> th ) $= ( ex sylc ) ABCDEFBCDGHI $.

Referencing norms: proof_reference/2ebd7e52e85f2983d8fb23dd8e40092256b2a47ed705de5e20f315e69f201d8e, proof_reference/9089616d826ed1c91b875c486f3275ffdf8f3406984e7b701267fa0700abf3b9, source_statement/cffda9350e5a357ea83e4ae4e72bbf78c24cbceae0b5104b2c4d6fd2b3acbebf, statement_hypothesis/712063c33a3da023a0e8db508c7894daba579c89241f4ffdd52d400248ed997b, statement_hypothesis/adba73cfbeb819d9c9933cd9e7c5108c5f348c53f8cd94195fa91a3da05fc7c0, statement_hypothesis/f4c6c5e1ed169c9135a4aeaff36068a8b7ceb4e4a1032ed0d69bcffe73b880fd, statement_uses/0ff5164add93b882d46b5e283f2f31948d779d5475a4c7a69fad2c9e64ea93ea, statement_uses/4afb2860dde65cdb96d829932d172145cb0dec0f5e33d38e1c0073da1b7ca8d9, statement_uses/57ebba15bb3c739bef0c25eb4f8e586e5de6f771d20ebde7d86c5278dc7dfd6b, statement_uses/7ceaa4480d83b6419cc4ef3e54a799260fee49ee226c02fdf148b198b0ceefb7, statement_uses/9887a9d4729a67f3400002446bef7e2c8ee2a9d0c27ed49550bf1b7761912c07

en · unofficial · sha256:aa0328dec48c30c0a00b76640a3c4a76d954cbb24a42de6173579acab95d18a8

a1bi.1 $e |- ph $.

Referencing norms: source_statement/bc5b7f8085dca498f27b3111daa99f029ea1432bdb03e57000ac5dea3b9a0159

en · unofficial · sha256:a02c41886b88cbfa1e79fcff0c112b4d52cd4b53a72bea0d7559363619b5965a

$( Inference chaining three syllogisms ~ syl . (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE__WITH 4syl / OVERRIDE" if you wish. Remember to update the "discouraged" file if it gets used. (New usage is discouraged.) $)

Referencing norms: statement_comment/29ac9d8cb3ee1308190ca126a5057c6051dfd2a720ca9425134a983f98375b16

en · unofficial · sha256:c762136240ed7d3c25b85c5fb62838b09dced68e6bcbb76282fc6f34c510a51f

a2d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.

Referencing norms: source_statement/9dc5a46dd7e960aedd7fc2a36eaf603dce345a3b027cba720e1f1f6897a605c7

en · unofficial · sha256:9beef168a995754ee074e6c2f8f47ebd6f45609879570670903e740327fcebda

$( A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) $)

Referencing norms: statement_comment/df8bd93437d4c8d29fea39d18323b136806bb3f120b046c71d85ff858706e34b

en · unofficial · sha256:641be9e9c503741962a2e6eb67dadc73ca22c4a5e48e4a23b6ccf657e1dd7a92

axextb $p |- ( x = y <-> A. z ( z e. x <-> z e. y ) ) $= ( weq wel wb wal elequ2g axextg impbii ) ABDCAECBEFCGABCHABCIJ $.

Referencing norms: proof_reference/112f3360d70ae7f5bcbdbbc97b13877043a526a0d6576786c968d45ce27a54cc, proof_reference/4abe5308d66e7a640d5e013ed21002a333cc4ca6984ea34b6e6251500f4b1c21, proof_reference/6797721f9f1de5cd23cd0ce5d4d558df28e9204dff3020b6e3ec593cbec55bfb, proof_reference/74a840a082bcc01bb1dd1e17fa6c2d66f01363d09c33eee794209a6b11343146, proof_reference/af988a2c6c45b14669a00599809c342eb990eb75c8c0f57e1d346ad4d9cf39f0, proof_reference/fb76ea49fa042ba9a3b6528f4dbd91eb88c9d148dadf085457357c41211efb88, proof_reference/fd92bb0656071c1cfa33699087244b9c954c6e8f4a9575bc55e8af405e56cbba, source_statement/46b9d719f5e85c89e8cdaecc658594b151a6c9eb069dc514007cfbb049f237e8, statement_distinct/39d3ffa29e337c03db2bea4cffe7d7ad4051544fa2233eccf9e5d0380d41d2b1, statement_distinct/985cffb3b5f4dfa50a37153cb0e691831db8a153c97615355ec47c95bef2ddb1, statement_uses/214ef8f6b9597f6335c840d6e131271a82f8991d442bd3f5b8e57274e7d86005, statement_uses/2409ac3da22fe72e2b6275568ae9f75ea19ace6e8c57f484a3d8e9accb264ef7, statement_uses/7aa1c03c14c963ce2ffcc64e8ca8de417dbc8b973630de5bf1a60f4cb7bdd4af, statement_uses/7be6eed1fe482cbbad7eba0a3800adf52d416b753709928e5b336d5d43620007, statement_uses/8d2bae86b236fffe2ea9fad8fcf643a5177b2a907d77527441ff354a48927eef, statement_uses/acccc9d2839fc480a9d8e223b3bbc84b1f2b0de9f025b9acc2c0c5446ba9377a, statement_uses/af6e41f7beec04fc1f445795dab5f3842a7ec38324ee2ef6d04865076d732446, statement_uses/d0a1b6b37594f3f1c7c65f9cd389c585312769870bfba26e7571b41e57fef71b, statement_uses/e62d45916c829bf2c02f2422caad624898c6c96bfd7929db51b549082978b928

en · unofficial · sha256:7741d8e9f3f1123e4211db8045b865e53fb287bc4337b4ba92bc6d397c4ab0d7

expcomd $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $= ( expd com23 ) ABCDABCDEFG $.

Referencing norms: proof_reference/491833b4c8b9f97fb91025fb59d5a53cd4f542f3118ef6cbedd503742b9995d1, proof_reference/88b2f951c572cbcc1c59e0eb1d9cfc4f46c71b7a633582e429fd068ca8f31282, source_statement/843c1eb200b66cb5c8f879937e95ec656363dd65460f0b475270a2503c161956, statement_hypothesis/ea6e7eceeba5a65420929bd66e2251e104a3b0328579df11eee2fd860e266766, statement_uses/1c278ff1c36b17c529961118309a48ade1aa7debf87923610b4a2151322671a6, statement_uses/1e6a9810c427bb5c75ea42c054cde8901868aa138c846540bfe70fa70ad414cd, statement_uses/25c65d5611604589643003581473881a006531bbcc61ec6d4235f9a2e2dfb89f, statement_uses/4028ad263baf39dd0edd3a8f76695effb7a000fdadaf77356f2046eca07af0c0, statement_uses/5a040f5fe1695ccef0ab1f068129bf97a67f17269bcf8cca8ec06ae475f5fd1c, statement_uses/74d33a7d6f705dfa60ac1a9490f13692dc8d76d462dd5ab34a4dde468a91eee4, statement_uses/965bb45d748fee0bc4d07068cade28757018fa6714a4971834109f9e5bc67173

en · unofficial · sha256:1c603187bdf5faf7e1cd3068bad0c46ab443dfe9a64e7e727483548df2553bb7

sbbiiev.1 $e |- ( x = t -> ( ph <-> ps ) ) $.

Referencing norms: source_statement/80b12c3680af6f58f581f32c2a3a6e8fa6d606a691145b29a8f7b3da90f88b72

en · unofficial · sha256:7fad308d4d3e963ee478ad44544e3259d449634963be2d01f5f801baba6ca4a9

mobii.1 $e |- ( ps <-> ch ) $.

Referencing norms: source_statement/6f9cf307c2531a46622bf4460ecdbc8d402c126eed52ba18445c22b730aba6cb

en · unofficial · sha256:e8614c59562a839c3efb3185099f0f8b9042ee23aa7d04b4c767124c0d5fa9aa

nexmo $p |- ( -. E. x ph -> E* x ph ) $= ( vy wn wal weq wi wex wmo pm2.21 alimi alrimiv 19.2d bicomi dfmo 3imtr4i alnex ) ADZBEZABCFZGZBEZCHABHDZABISUBCSUBCRUABATJKLMSUCABQNABCOP $.

Referencing norms: proof_reference/058fb6103060c604a2f07f80e1e1306950cea05169344fa4560e67313c9e7db7, proof_reference/160d36b2b7b90c0a0644a6d1d39e03ad02fcc604426402643771670d3845ba15, proof_reference/1a13fe3d08c29537fd181fab763b3ce8907d57f4261ee41423c21809ec1d5055, proof_reference/31a17e4938bfb1d3c2618cdc49353b396027ab4ad17123d3538023ae0600009e, proof_reference/3bb7042304d809a1afb34e9f2adcf3e9abed3a34c6e3810460ff33cfa1104d8b, proof_reference/3e84b4c02fb68e799e6e5b5adfe3a649f7d5548a3ea8ac7c83196ca8124327b4, proof_reference/4115336e809869a791a35ca7d547a12bb56e4f8c6462486e17ea252088f605d0, proof_reference/4e24cce7ee6431caf8ca65e12aef33b84ae8a83bd9a0fc7e6f224dcfa2832449, proof_reference/58ddf0478960884e2f6afafc13615308a7b96fcb2032ea3cb9712409de626865, proof_reference/69bd6fc0e14a25f923a1dab739356ad3c8bbbcb03f10a596dfd865719ae2bf58, proof_reference/6ecc67d0bb4dd0fd8b89ca902a5b182e8ad0a58ae93e137dd59a51e4d4f25280, proof_reference/855cdf323c960a79540b563415e865965da48adbb53ddc0cec9c4000e26ac366, proof_reference/9f6b7cd56fc34c660ac046314677430a92149661fa3ed7642e9c06167039c473, proof_reference/ce0e8bed778b18365c13ec81d234f9d4572b7dce16719703cb56e63b9a24a413, proof_reference/ec6d9fdf4821277cb8f024a5386f5a03db2030bf73efa9a9dcbcdbe74cce28b5, source_statement/1b2c6ce6f1766bdf94bc71940aa5509203203f18a678b8d971d215a44fdae84a, statement_uses/004c2a71502642036aa6afd1fa8d7bfe266cf5a8620cce8a48c473c2aef2add7, statement_uses/05d180b49622b4972e5b8e0f886169b83d3ef63c4681947fb2f70b92124cc3de, statement_uses/26681fb9e53c2b0ebc67f33c86d125f5c0b0eaab1a2ac3f8a34344e3e6dbb9fd, statement_uses/30e422943ba6c5153c043c209b1e4f03520b19402ca6f47188d756a4bf4c658d, statement_uses/33958555c68d9d62dbde3765317ed441250ff99d06d227329af53ed3089fb49c, statement_uses/517be0136391151a776ca8a515c43611bcb3dd17cfa9a67f30a952befb5d1a1f, statement_uses/88c3dd1dae489dbccab3742f202afb3fb09ed76403c0679ab55ac61a3ebcda79, statement_uses/ff0d128e2c74bb08a043f6562b25ef335a2fb21e02031d9145a22a30193de605

en · unofficial · sha256:d9e5a0a0bc5250cd7d4035d0354f65b1e4de769eab0877916beed306b9245e8b

elissetv $p |- ( A e. V -> E. x x = A ) $= ( wcel cv wceq wa wex dfclel exsimpl sylbi ) BCDAEZBFZLCDZGAHMAHABCIMNAJK $.

Referencing norms: proof_reference/13848491cf00999d5e9cf98f42150c9efa64db6b73caf33c132accb0105e33e3, proof_reference/2710a45d51c403674d147c294b4bc70c96fc08d3c5e1384ca0f8edeace760f33, proof_reference/3935ccd33b40d1f045a1fb753d5ffd988be64ae0becfab31b34e36ce92d05c87, proof_reference/7525002d3351f086e465756338ecfaf6324e77ec796dca87e60775e5fcba8fe4, proof_reference/96f5e0c272fa028600f011ed89b90fe91a4ceeccd0b4d17a13af52da7db865a5, proof_reference/df86eee5f196afc4cd1f03aa4738d714905d5aa25a92eb70a81bb7bd8a164604, proof_reference/e837c76a35b73b80cf0456a74407e415cd42b67a9db65e3ed2e0ba233d65a259, proof_reference/fe23140f2a283bb29a4928a10cabbae103b80263e5a4f83d32f1054d79c6ca17, source_statement/989cbfbb437ed479363e34f61c62ecaba96e400b88643fdbadcb5f983ea51143, statement_distinct/3d3d8f0ca1263a4016ac7a7e33e68b023bdea7fac6187519496219e20fb12ce3, statement_distinct/ccc9e793854d2a8737290efeff31804c1c6d3cdd69ce72dd7e6b62b54901a174, statement_uses/24c384d86ac393e42fd2c445a039574a287b78ce3fb777814aa9197f31ec7c8d, statement_uses/356c3ef089c4e020b3a81e68c26dc7d2ddfdf80079cbf461e3b259afc66c5d26, statement_uses/39ebb971049ed818fb63ea94c8fdbf7d41eed4512071009a1d2483d425aea2e1, statement_uses/3a4bd42a3e897a102a578d052c511397231639eaabf369f2ed35b28fe47c4570, statement_uses/54c19f6a81fc03c7a9b15b5c190ae6ca36009ac8c488f67567140f139276619f, statement_uses/5a0c439a69dedc092d2047aac9b38bdc838e7cb5fabc66689f238417b5baa565, statement_uses/857e7304cfce1f2a93969b15b8c32dd59ec193a88830a009faf58c54823847cd, statement_uses/a3a1cb532a58e17cf136a6474fb57598d016858d69a3a9a68555ce7a3976110c, statement_uses/e5b26be8f162316f7c225ddba6776683244edd9004468d74c9785184825cba1f

en · unofficial · sha256:147016ad92faa01055b344a2ca8810d57e48edbe26215a6f1493fd8f19544ee8

con2bii $p |- ( ps <-> -. ph ) $= ( wn notnotb xchbinxr ) BBDABECF $.

Referencing norms: proof_reference/3718d59f9d927aa01750b226812c96f1564b49bd7dd1843ba84302eec4e54f0d, proof_reference/6fd95ce9cab8e846c4304bbc0ecb3fa6dd6a19556e7899bc889c133153a45d3c, proof_reference/8922dbc0743fe906d3c9627096a63900fca2e1c1b2af842286b3bf38f5b5ddcb, source_statement/0b52d335842c0799a5055e057f73237bba3795c56d364dd98ca3dc22dff02971, statement_hypothesis/873a4182b58f64375393bead673c38ee5b62ade5793cbd3af42039c779336f8f, statement_uses/11706378fdd7912fbc2e55c4045012dd0ffdabe00b790109ae04edbaf71b3514, statement_uses/12c75a2c487628ee9b6da9484f291754c3a7af54cdafb06672238f7fcb5e2343, statement_uses/9cfa4ab6826b0e192fe578b65d97224a7de8a71428b1ad5822b107a2c1a9cbb5, statement_uses/d1073742385d25fc13fe7746d9e205c024ff97e28aafe0f99ae30ef2f9d15d05, statement_uses/eeac4b5842f0ca83762d9b77e8b691590fe3b1e185ac6c1b6aeac7dba293a4d4, statement_uses/f2e0d441784824dbcfe9a4bb9d03d36c158cb892e2efb83f506dd6ebacd84df2

en · unofficial · sha256:5b99109fbf1914abd26539f8f098c51591a399e1f2b2184580a215a9d87b7109

$( Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.) $)

Referencing norms: statement_comment/63dca6873b9cdf2d7b55af8b1636921a8a6e75e68781f55f7c2c49b4ed507c18

en · unofficial · sha256:3a07f14a03dab6cf6997369a1b84d2973a9288971e12e38446d33d5bbcb4d64a

$( Deduction absorbing redundant antecedent. Deduction associated with ~ pm2.43 and ~ pm2.43i . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) $)

Referencing norms: statement_comment/d56d476a62c6f4630ced545fb5133b6727367bba249c112f1eeb849c8227b40b

en · unofficial · sha256:4a23496dd6174c267855f022b42128816d1d6e08e0b1f4478372dcff7bb4e448

$( Every setvar is a member of ` { x | T. } ` , which is therefore "a" universal class. Once class extensionality ~ dfcleq is available, we can say "the" universal class (see ~ df-v ). This is ~ sbtru expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.) $)

Referencing norms: statement_comment/556300e6e5553ac7147c6cf0d94cfaec4909fdf205fa42ff1e87b47e37f54de5

en · unofficial · sha256:037ca8cf63094ecb879181b8216d7f5ee4fff6139ec23be9523dac33ab08169a

imim12i.2 $e |- ( ch -> th ) $.

Referencing norms: source_statement/88188c4b91d5f67de61ca5e292e70ce5a40a5da9fee58e3a88e8cee50dfa4d6c

en · unofficial · sha256:22a9b17fc0c1edcc9f454e1bd68ff28f2ac425a593a091ab0403fafbe41589f8

3imtr4g.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/1c123375323e9f283ecf707e8b492d662843a8336682ccb5fa4dc67b33cfc0b1

en · unofficial · sha256:38d54b03757a7c854fb672f80b2272f5603612266d94bf322599d1fbdf6ce59b

3bitr2d.3 $e |- ( ph -> ( th <-> ta ) ) $.

Referencing norms: source_statement/15ece28defd603d114924ebc97180a605147da10bfbdd2ee63f1edc09e59789c

en · unofficial · sha256:89c2b7fa74a526e09cb17c38a58036704758c9a15dcf07f49f8200921c9e7147

orim1i.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/590f43626881749e0786e1fcba6e594759df021c731c527842667fcadffe56c3

en · unofficial · sha256:755919aabb0d7f30c7ba32760c930efd04b0cbcf8919583a1793441bd4680e22

pm2.24i $p |- ( -. ph -> ps ) $= ( wn a1i con1i ) BAABDCEF $.

Referencing norms: proof_reference/79e8e3afc6e4439233524c7f8545fba208547589b595d631855dc80429916fa4, proof_reference/a2e329ad033bae9ea63fa5bdec5ec346bee68ff900290a8c921e0cacb41147bf, proof_reference/e103ece4ff636d5b4c2d7fb8f1334a6bbf7ebea61f059a685ee88fb5d06957d0, source_statement/6bb28f7f14b9aa948239b097c03efafea8d8c506728f5ebb17078a7f6fcd0c99, statement_hypothesis/15be306e8199519170fb32640f8050b43cd0b859d2e7c814b64652db5e2c94c8, statement_uses/07c4ee5b986ae17f8c512272d952df071d810dad90a0ba2bee7a8c5c6d7013f9, statement_uses/0828dc1cc7210554b35daedaa3ce40a9efecbd5e8949d621ab2574a67b616f5f, statement_uses/262e3baefb8ad70c68d76fc266ab941f762c43cc16e32d5b6eab3fdcbdbaecff, statement_uses/6c6d4d139ae8be1a13d9c12bebd9267ee3e5695043c484267b40bc76443e5ba6, statement_uses/89714c81a0625ee241b09c1c5b6da4b07b4581bcd74cf88980e122af52178cc6, statement_uses/d3ef20fb82fc7b0ef1dd34ac6975b5d3a26a7a582cfefabee9c9512ccf333423

en · unofficial · sha256:a105b79a57bb0172527f14d64a81690280d37fe91efb318f3fc7e6d1b42a0877

mt2bi.1 $e |- ph $.

Referencing norms: source_statement/de6c343b1528de206a0085e5afb12b9657e5720eadc82e87e74f0e8c029f407e

en · unofficial · sha256:8108b2b506218afa8191ec8a449c9ad24cc3280eb91c1ded8e5c188be9a13202

syld.2 $e |- ( ph -> ( ch -> th ) ) $.

Referencing norms: source_statement/8533388b482d3bce9c538ece3907ac8bd4057f3d95ca7e83538306bfd2fb3c54

en · unofficial · sha256:83d490d4a6d686976224595b3817edef53f45d064a4449bd889ea04c1a88923a

$( A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.) $)

Referencing norms: statement_comment/ac026692ff465ebf8c9ee2cd6054838061c0e9ee9dacd42e2aa6810a676aa03a

en · unofficial · sha256:323ef4e21868eaadb10527d77498128317d633dbf1fdb0fa62b1b849692b8977

$( Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.) $)

Referencing norms: statement_comment/31d84c01c1f24cfdacfff84d4b6838cb473c6508a189678e3b0d0ec0d997bb43

en · unofficial · sha256:bb9b3a745d5a0c44b4afe998076c90d9915d26f1a7ea75e19cd4d9f942ee54c8

syl2an.3 $e |- ( ( ps /\ ch ) -> th ) $.

Referencing norms: source_statement/fe9372a0f5720df8493d3b3a5c948bb3721ee3b2972218b9fdf170470d88443d

en · unofficial · sha256:b2fb03d366d9bb94e25becfd3ce7fd691e81820018a5d78274d7946ca9ea1296

idi $p |- ph $= ( ) B $.

Referencing norms: source_statement/f6f0e25b704b63912118d6b0396b6e0543e1a7bd6c429dc2eab191fe4facf9a0, statement_hypothesis/0e1f51b1366577f0ca7c927fa09a25476b0a9d79857636b5f31a084ddb9b7fbf, statement_uses/817cbf711d91d2bf899483b1871309760784d538a43c3130f7efe95432bc102d, symbol_name/b3365f656712b959d1b91ba87d29b83b6dd181c7d47db8e6f71fc392ab69015e

en · unofficial · sha256:76783903a347115f0b1cf8d806126c397eb88dec58941e280235ec0fb78a91ed

bitri $p |- ( ph <-> ch ) $= ( sylbb sylbbr impbii ) ACABCDEFABCDEGH $.

Referencing norms: proof_reference/107939b2eaf88402b495fd379d95710b34993cd71abad4ad5733d7e27b343eac, proof_reference/2470e69b8b87cf884bc68c942eedd1079cd0af479f1161aeafba39337fb44203, proof_reference/b412655d7472930ef70f2ab87e1aad9002278b72a6528223c3060deff1ca7ef7, source_statement/78fedb6b156a6af09320b9277d1cbcec13d03aa526eae4c492db0d4ae5c4258e, statement_hypothesis/5048eddd4fed2bcfe318357a12f54d227114222a4a79019dae0be6399700d4a6, statement_hypothesis/c7fdb2672aa189e97ea0385d81d89779fe04e71a7e60d34f7fda385c19a4aab1, statement_uses/19eb77cfe47ed604843c610cabd1e5bb4ff57be73d4253748ec6537a8ad43de5, statement_uses/3184b0a1da723fb4e3e30f8888c2863bd4e605837948e9657cd456329b8fd8ce, statement_uses/eb1ceaeba580e95a85cf68adf84c3e8ea8548c9dbc534f61858850db9ec6d746, statement_uses/fa37f3006800f2ffb7b8a64123669b7b0b8cbf5e39536cfc740295cec26df01b, statement_uses/ffb5d29ba29f6445b8490250cf11142f301508eaf5cc6a59d5ffd2848cc761bc

en · unofficial · sha256:e9736ad42fafff7d18a7db131e6bca3921fd6027388f9483bdea8730ce23686d

eximi $p |- ( E. x ph -> E. x ps ) $= ( wi wex exim mpg ) ABEACFBCFECABCGDH $.

Referencing norms: proof_reference/10755cc5a60228e3016212b23d2be4e8791406f5a122c7135893886c6f021b23, proof_reference/5384f209beb49d43b67c8b9556b83e682daea3e29e00f43f8ab3568146695b0c, proof_reference/b4caa9a1743c1568f901f0c7fccf33b91d248cf8885e3cc4e118efa3bae0d283, proof_reference/c193e94a6ccda8d2dd80864c2a92690f6af3ef7cad2d933aab8afc97698efb9d, source_statement/31f07c0a4f4f0d72f15fcfee732e0439ba3699721d3e5d0a26f6c3d071bae672, statement_hypothesis/67b1751c0b00644d86554c2b24d6c4fe74dbde9213e80f7320b484037ca56417, statement_uses/5ded08b92c07a627da9977360cfcc88ac4000cbbf5bc202dac3f4170b8331e8d, statement_uses/9753dcfa982f0b9f1d083ddee97af049a37bc3bc841c412d4ffb8c3ae534c153, statement_uses/c1f2668c18e64d2e43c72cb5c9f86e34a177f89bdff4075a85f4043c68c3aa29, statement_uses/d22a6a684913193c4d46cd1456e4dbcea42739830f9c836b8b39c1107a10c863, statement_uses/dd9721b947c7b6b998915ce0b6898d12d6e9153c8d0df5aaf1ecc4f053e23661, statement_uses/de5b662191716cd77565509955894759631a84796874068d1b7cedee6d206f05, statement_uses/f86f6f70b3a59325b940752f92dcedfbcdd3d4c19777728ae81750757687eaf1

en · unofficial · sha256:82191b9b0ae5b00a27f3b273b6343b96a027bb9c8c586eac57484b5e5e5ac124

$( Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition ( ~ sb2 ). Theorem ~ sb6f replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem ~ sb4b replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise ~ df-sb . (Revised by BJ, 22-Dec-2020.) Remove use of ~ ax-11 . (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.) $)

Referencing norms: statement_comment/03c09ec766023c72fd381049aa344a418903a33e236c67fb29fe28dcd1f16457

en · unofficial · sha256:b8fab383cad78631729b7f803ab3bf10cc4564a4e8f4fb98d9394a76deb667b6

$( Double deduction introducing two antecedents. Two applications of ~ 2a1dd . Deduction associated with ~ 2a1d . Double deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by Jeff Hankins, 5-Aug-2009.) $)

Referencing norms: statement_comment/4592811294aa3de92098c451e9105e2ab81929ccd727d4ffff4f74ab7b93b14f

en · unofficial · sha256:96e10147a769177a3dbc3d0497769040fc745a8e92bb0b800aa07163044421c3

$( Universal quantification of negation is equivalent to negation of existential quantification. Dual of ~ exnal (but does not depend on ~ ax-4 contrary to it). See also the dual pair ~ df-ex / ~ alex . Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) $)

Referencing norms: statement_comment/7ee8fd12d3036173b389776428076904ff8e07c99c0ed6afe3e118e08e74bc53

en · unofficial · sha256:65d0284ad79d52b1e4cc76627d85b44f88168840a4cf3961452468db41fd0629

$( Deduction form of ~ bitr2i . (Contributed by NM, 9-Jun-2004.) $)

Referencing norms: statement_comment/0cfc9b29e955ce84035c840ccc65fd82d260e355660cf4fc3aaa9276f6894aa1

en · unofficial · sha256:fde50b8bcbb43fe4f6a765ea6762795a8d285e0129cdafb3ae11e8541da71b64

spimfw $p |- ( -. A. x -. x = y -> ( A. x ph -> ps ) ) $= ( weq wn wal wex speimfw df-ex con1i sylbi syl6 ) CDGHCIHACIBCJZBABCDFKPB HCIZHBBCLBQEMNO $.

Referencing norms: proof_reference/03c0a5ff2e1d9219c0985ce959e9c5dfcd1dc58915777bcf213605a57021bfc9, proof_reference/527767fd46a04ce2790c6906df1e194867f60b9723bf3bdd0d9ca81e2ae31d2f, proof_reference/558691f9019e6f544702821ca7621a5df2049b58ac0a61633aabb940457fa5d1, proof_reference/9b37791deba88f53cd512216072a44ab98979349df40c4626dc9ca3462b6e18b, proof_reference/9c5e51c84b4865e22cd71a1d3310c0bebb0f2a38565577fe2cfb883cc6feeb9d, proof_reference/adf51b72e8eb0cce3749daeda0cb17b85ca5a179a91b6cbe214a2c2213f09e7d, proof_reference/bc97a13041c9e4522bfa6e18fee70da59a46864cf9bcad42fbbe76de64e0e403, proof_reference/d9a66d85c608e577345d91e770db9ce6c18d66ccad79c8f603e9d4816a49add0, proof_reference/e3c0398fcdf7029ab9cbcb8b20fed88e9cdf692ebc46efe1530a13ba285798ef, source_statement/71aede879fd68d0e462a573746f93561357a48cc8701c7304dd783d7af51a000, statement_hypothesis/8ac431af3efc5ae5dc5a7ba10d4c730671759bb568babf92292852b417bfe609, statement_hypothesis/e2bb1b6f7ed798e95fff0ccea8a2b94d70de082699419a5ec47f89adefe34cdb, statement_uses/03260ce1deaa7298410be64e6f6cf464f651db0d203c66d7713fc3d2439ee095, statement_uses/1b47d8ebe2a3a5b815f7b86123f073ca79100b147f6d06a0de70fc6f691d90e5, statement_uses/5c8a671aa349936492b37193d01633f7824affc9bf68981f08a1c09360829d25, statement_uses/5f186121df0430b48a524ca5e80e540a2f61399ecf3bd97fc9b86df6a57a630f, statement_uses/7e7ed47c9f48809758ed7d638e217b75a5cf49dd3600b9ea3312467f748929d2, statement_uses/9b559de49a4ceb76eb556db7e953627e2cb2b91a0a596919305b08bc1ce664b3, statement_uses/a1df28a8b2a383cc5d1dce4b7bc847277dcfc735e0e0776937038b1c6152c2fa, statement_uses/acc90d650d67b303756e6015017bb058ccee4372b780ff132d7f8d46165d5fb7, statement_uses/bd8871e1cd764429afa0e1bc40239062f0894806edabd34368d2f47e11c310f1, statement_uses/f3127471ba199f80ba2cfdff7c770e16c783c01f0f48fdec4cb631b3b5f4dbca

en · unofficial · sha256:a5ce1cc8ba8c88766b7ff8eb8c133d9d0cee3ba211858d203d73903c705e6fb7

pm2.21d.1 $e |- ( ph -> -. ps ) $.

Referencing norms: source_statement/c6c77bd2929faaaea22e9a7c3985801181498a0bae9c52d8acde426d6e741a99

en · unofficial · sha256:328a6c8b63cf4f7feea490d3356dc1f6710b2eff9208694781aa431676e42604

$( Deduce that ` x ` is not free in ` ps ` in a context. (Contributed by Wolf Lammen, 16-Sep-2021.) $)

Referencing norms: statement_comment/e657cffb46c37c285124db61fdb46cff5dfadd4f2c7d78e158484cb6db0022a8

en · unofficial · sha256:83065c1b966441846d710554e7319aabd975565aca73a2eddd82eb445ea87994

nfbid.2 $e |- ( ph -> F/ x ch ) $.

Referencing norms: source_statement/ac351b21063c94471fa543f537ecd7f5316c623d02840008165bf699d1bbb59d

en · unofficial · sha256:1d60b984973ec809056bac82d0a77fe8007873a4a6d0202d70110bf41f37eb75

exbii $p |- ( E. x ph <-> E. x ps ) $= ( wb wex exbi mpg ) ABEACFBCFECABCGDH $.

Referencing norms: proof_reference/24cc0d750cd27d24b8d1e178da4dd92aab855c9c1d45b6a161cc7c9919749e4c, proof_reference/3834d6348d8979ba1301ac0e23f76ce668a808c6c0e00b95fa8c6eb2596095bd, proof_reference/9744faf275005dc332305be60008a7ccc7d5ce6278b1bf09aad562afb68b53b6, proof_reference/a6c1d20acc4c3ca9524b7c07c236f80326e64f702736d65a0906907e719ea056, source_statement/d2f1a222fbfefb7846d88682f746cfd9f3fd02cf4cbb4c43bb312d9ae9a3946e, statement_hypothesis/2053e1970b42e919df6f8911f6bc039e220cb95d72d5babff76f98f425f65e24, statement_uses/084c52a5dfa5f69afd9ee636b3bef57ad4a042fc0900d2d69ebffd3cdacbccda, statement_uses/0a9827d7084b78b387b19b512a164bcb4615fcff700c1e478f046dd798a2264b, statement_uses/3fe77b64c836e5170d16b10e6a31e57d7d8b25894fdea469921c8b9fbea730c9, statement_uses/6e79e75537f19cfaba5774fde55d32bc9eacd5f7a84b907ec713c0ae42ce10d4, statement_uses/756249c2713c1cffff4a72775bc3d0d24461947ed2478fee45d4c8913825cfcc, statement_uses/c121b2150f6f727cef70f15340414caa4b5f6c0b4a0e7768e616d429b8c12814, statement_uses/e3e522c31230e0a4f19a0a53c0ec26911d0d58ebbfda176b375441e9af767689

en · unofficial · sha256:da11f0ae696ec2887c16e872c46456d5f5f25f61322be53a67c79005e436d412

$( Extend wff definition to include class equality. For a general discussion of the theory of classes, see ~ mmset.html#class . (The purpose of introducing ` wff A = B ` here, and not in set theory where it belongs, is to allow to express, i.e., "prove", the ~ weq of predicate calculus in terms of the ~ wceq of set theory, so that we do not "overload" the ` = ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the ` = ` in ` x = y ` could be the ` = ` of either ~ weq or ~ wceq , although mathematically it makes no difference. The class variables ` A ` and ` B ` are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See ~ df-cleq for more information on the set theory usage of ~ wceq .) $)

Referencing norms: statement_comment/623c9c7a4505d3f6e9e0addc9e04d1c4998035eeaeddff4801d8d0057dbe3718

en · unofficial · sha256:519449d2223434295c81930ccbcab45f0b19bff41abc1b62e8a5c7550e5017c3

$( ~ nfv with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as ~ nfimd . (Contributed by Mario Carneiro, 6-Oct-2016.) $)

Referencing norms: statement_comment/5a6c5b6877be265a9644af47f8c0ab76a2d8cc0f46a643ffaa74c39d0e07c509

en · unofficial · sha256:62e88b3439bd654f5e34fd93c04c3e479fab24942942d5f589a347ac3fc63180

nsyld $p |- ( ph -> ( ps -> -. ta ) ) $= ( wn con3d syld ) ABCGDGEADCFHI $.

Referencing norms: proof_reference/3820e439eb762fd5642a7231558d0636f6714b2ea5247563ff1e4df766c69cf2, proof_reference/69433435c419b59505e984aeada3a68047fc60619393b314006f2f42833f50a6, proof_reference/afa1a3a110e5b2ddad499874941e5ea093fb5409cd227f612798e2653ffc086d, source_statement/91fefd2a347fc76364bc54e91a69af2bf11930e1c72506662aa90e71a6be8f5a, statement_hypothesis/0a8513f6fcccee5a508a6e885cb77e4771fa09d5cca60a2f62699c17c256b0a2, statement_hypothesis/8a4003be7fed404504aeae9a2bc40962867d82b82aaa12895565fdb9c5bf3845, statement_uses/0a222ae1ba2ca14296cd8caff0db076ab4668ccee7c42777426f39ce5862412b, statement_uses/15eda9bf195af3597c3ffd5bb4d8e68a08c6f28b14eb907bc0b36a04f1213a03, statement_uses/30217fa870693e72af183b322915164e5bfd72e1de07913d8b4cba7a328b69ca, statement_uses/4f52cf78c1af98fad20d425e18eea1bea71d900468e7374245585d199b640d67, statement_uses/7790de8459bcc630e6e2cf255d6d79baf59991fa273d6d15d7080f7dd3557877, statement_uses/95be861d757ac7eb071aa4d6213860ed9134aee5db32d0e3a72e03504c1a55a8, statement_uses/bc59837b3912f5bbf284c67b54c19d523788b3bde094b54bb22df38da25163ac

en · unofficial · sha256:9ccd604d13925eee0b5b851e0c6729625603654bf069b02f5c49410e2aec1b63

eleq12 $p |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) ) $= ( wceq wcel eleq1 eleq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.

Referencing norms: proof_reference/3076ca6abbd5ef8a1d4bcbc96cb3ac1f1f12f20ae80a35a00474531c820f87b9, proof_reference/5551f0d9c71ae3a3d96f7d832169fd2c4a14cc26eb386d05d4f7f7f8d62eda84, proof_reference/774cdbcaeaa4aacc0e860246e2401461519aaa304855ec1991e211bfaf0b1ad3, proof_reference/97af8790ea7de283e7a6e46dbddfac61ce54c9e62e31ab4aa633095fd695e16c, proof_reference/b7e2839da3cae6d1e7f3efb2254e0a616619817d6b2ca79cbcb47ea256c1781a, source_statement/58a4dc2f9267535fb039a05989f7a856d0f93afa5c1a5c5f86b4ba7252cdfb6a, statement_uses/0d812e74867ae69e8538fce7cfe091b7d86ee9218d730a9e9e31e3db37b6e097, statement_uses/0ea2fd3f382fd857180f8281012bd17fb6f89625f5602a0c3a8ac662e16c6a92, statement_uses/4dcd0e3db1069c43f3c1e349ff2fed1a9f24828e2361f33ac35d979c91744965, statement_uses/59be40f0e501ec49a2d7c8706a62909855ca952f0e134484b7241168843e1c0b, statement_uses/6a4df5ae7d078c4c0f904639540b4e8e9cac1dded0be38fc4e7ef150399910e3, statement_uses/6d9f9101ec9aafe47541a1694d0bc561b4b92e76644151667a6e839861c28eac, statement_uses/6dbf6fd91af52f5a8f4b38ab106ea4ddbcba196a43f7801f5fca601e873ccc2b, statement_uses/8ddffff96769417ec847d43f3a9ef8e866b50195806a314465b6bcc41456b9b9, statement_uses/b96850a18e94c95747d8b3838c021dc0f6f7d6214e42fc0d10b1cbb04c011217, statement_uses/d4154c4a29dc652fc4b7c8c013fbf7356888ff354209482d9fc43292a4583bd5, statement_uses/fc10baef9d822a4da929afe4ffd5dcdb0d73073d8357303ebd1369cf05e178b9

en · unofficial · sha256:9dda24791f62bbb48337dff222829bc6ec7f518f8a0288f3630770cc673d8555

$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) $)

Referencing norms: statement_comment/428808ea066576cca2f071929ec77bea66ec8c1e00ba7f42a629722df4a3b867

en · unofficial · sha256:425b51ac29501772018475c4ac22253ec1af8f327bbceaf503cf6c8892435157

$( Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.) $)

Referencing norms: statement_comment/5aa8a17ad9923aabede14d1e892a488a28fb561711c0c17ee96e3c8f05fae869

en · unofficial · sha256:3c9518bc10194051a4fa3c7789b8340d8adb7ecb8673908b9ac2c2fd65b9594e

$( A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.) $)

Referencing norms: statement_comment/3e378e97c75e8cd16d2e2d4b3705eebac4630c38db32c53a1e052374b6895319

en · unofficial · sha256:491e4be8b4b6d17482b712e4ff3a981fedfae84a7c002376e845921d3c811e34

a2d $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $= ( wi ax-2 syl ) ABCDFFBCFBDFFEBCDGH $.

Referencing norms: proof_reference/15c686d7de052777f609796d0e9c50d388cb534b4f161d87dabeeae410e563ea, proof_reference/8468ee8e97c3659a3ce31d432c058f7265c83aa5b564086558682f5ac268cdd6, proof_reference/c13aa3ecd55affe316b06845891e120c3917c7c2bda790cf484da61046b200ed, source_statement/a33171ebfb85400972021563af52c460e75be349633a2d68ab54ce9c9848d81c, statement_hypothesis/880fcc967b6318133edaf30b08b1a1be8a931543adf8b3351155a871a47ef4d6, statement_uses/64f8521d0743705e52dfc055f608a017375dae4a5a336d42fc5f58b9db95cd7a, statement_uses/c039b061259f285c5eca86d3d21e7cf549e3b6a398a8aa556d3d9b1361ee435c, statement_uses/c1d68fde15a7462c6ce1812741017bc8962c685ed78a58c3af89f1f9011ae8ff, statement_uses/ca8abc9edecef561cc02d68fe2d0f675a08c08a3f2a3f1eda0e4ff439a37f7f1, statement_uses/cb9bd3f88694198279e865ad953eaf2c4f91a58b8511046c795e4b6f36cfc0e2, statement_uses/ec59fcf45961bbcb6d602087ea4f0b49f9ebd7dc20cf9cf42ba2d461629f3e15, statement_uses/f7f846c7665cbe1e44bd1718ee01dfe6c0f1f88f731dc53c3c2063a69d80b955

en · unofficial · sha256:f37566345fc323c380e5b0d9a1fae6ca616719684a2702644ad8cf30cd837d53

cbvalv1 $p |- ( A. x ph <-> A. y ps ) $= ( wal weq biimpd cbv3v wi biimprd equcoms impbii ) ACHBDHABCDEFCDIZABGJKB ADCFEBALCDPABGMNKO $.

Referencing norms: proof_reference/0da691be06700559df6dc3cdc6ca25670d26520a9f0522fa7df10467c6a3b209, proof_reference/180e5e6f5ba49f5cecde5c4168d457d0e5b1ade13734ef126decea5a9def78cc, proof_reference/72085bc7bc4e8fdb5d5073294f5a1efc075e32ce5a108b16362748311b4e78ad, proof_reference/97d1c734c2fcbba624ed9298727c2cedc863ed9ee530181a6892c991fe32483e, proof_reference/b58c2e9da05e4afc14aadef8fdd7e82f108ea76bd6d4b57a09a2bbce638eb510, proof_reference/cde4215ffd802c6634740b5545c03f4c26fd417414c52231df9c0edd72c47b62, proof_reference/d8d108d91b42e9287c1bddc356c51c4ed421889d8ec51b5ac18a79e4d13b925c, proof_reference/ef3bc449eca34146213e1d9567f0e01a800e0ee797f0c757da173b35c95e15f9, source_statement/092cd7ff01de5b23458c9f3180dd8995d101b494895abfcb5b85bde95d8b70bd, statement_distinct/b8c425e155665cbb84ffdbe2d7a5457a930ab52c357b1f2d5bfc2486ea4c16c4, statement_hypothesis/ccbace36d0a174aa7e0395cb401f37c3c431cf4d4ec715891edac841d09aa1ef, statement_hypothesis/e9aba4535f57cefcdf12197873eaeb5f91d3e3fd3e5eb20892b4118ad8ee68b2, statement_hypothesis/fc9c1a38b06ebb27b7c93550ded4d9f0ca45fa1c2cffc79774691a6e26e71d4b, statement_uses/042c366391343ca4abf65a51b9f18fd87ee8dcc7c2058341bd8912e3ada7dfc1, statement_uses/4c756fa5d834069496cc1ffb0eae953a34523f7999b96d7d39f7dc7ba76ad3d8, statement_uses/c0ef720b81333b90bd3aa02d33cd94bb93a322b7aa23c81a659608e3527dc9eb, statement_uses/cc6dad6b3931f90b2e23e6e164bfb93a785860869f4a6706ddf529f5de666d48, statement_uses/cf78ad7639b9cc016cd2572d07d9db226e8d40d3fea44d72dec2b8931c3f988b, statement_uses/d1ed4bd3bef6fbf27e960670355b97a51a46d89e45b97b48c667d9b78603d73d, statement_uses/f04684945e3f0401ae395f502f88b085fde3005c6d43834208f54c8fcbb7b13c, statement_uses/f9f4d6235c32e447a303d074dfb321b588062be462a3bb39f4dafc7cf5b11ba5

en · unofficial · sha256:135477e82369ae2fbe0d3fbef393504ab874d9314611dae3d2b374a383ba535c

$( Extend wff definition to include the at-most-one quantifier ("there exists at most one ` x ` such that ` ph ` "). $)

Referencing norms: statement_comment/f7e6a5d01dbc677d809a2c67a5a5901afe156d45a0bd53ff7cf90c1d14786507

en · unofficial · sha256:0548760079a8021b467eb8b6f4a86e0dda99012769670893faea47805fadfb59

wcel.cA $f class A $.

Referencing norms: source_statement/ce564dcf8b054bfe125f391ac82ddebc4836d1a9fddbb3c96a432f13cca3a2e8

en · unofficial · sha256:2da143b4be934cd3050bc4294a3bcf221ca7cbd4d80a316e78a5b1c088137399

pm2.86 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) ) $= ( wi id pm2.86d ) ABDACDDZABCGEF $.

Referencing norms: proof_reference/19e0fa309ae19ec97374c3e7e13ca6a75a4cb4c4d0bf4f48134c1d3de7dbee3b, proof_reference/ab550ca98139b76d16adff4f698d55fb71c091df48cf2ddf45b2f5d6afc477bc, proof_reference/cfdcba4781c603895b3bbdb18331aeb2f411463e9a7940eea00224a5b202b182, source_statement/f637e22e87ff8c3176789aea696f963d0676fc339d2bd9168847512f2f13033a, statement_uses/1ecb8a440b2b309aa3f1a4b545d603e706f8bee1e6c2be6c808351556a3c5d54, statement_uses/71b8c18c8c31c10f067d87319ebf796609f88a793af30d2419848278fe2eee0f, statement_uses/a2f396fd1f0669b7fbdd5e40d343d1cebf1216f5792bfa8c2fc0b96bb0a4b171, statement_uses/ab3f779fea908288727c340ca4e5e420f3350334ee2741b68bb9b03426db640e, statement_uses/ec25ac8093cc2ecbe8f54c394e75ed1c2339b6a0449b1a0c4796eca5c65ea0ae, statement_uses/f83eadf64b55afd366db1312b2f917a9f7ba57ab3ba559ed7f2b453395f4ac16

en · unofficial · sha256:5bca898441369f0b07794d9f25f79000204e449b90f269b919d996acb7245894

imim3i.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/fb1fbd83f1bfd7bc324981a57cccb81e06576c952ea3f5d9117b51eec54bff56

en · unofficial · sha256:4c81286e0ac7b5540b67fac0e1869ada77222d222cafb9fde9bd664c180ca3dc

$( Triple syllogism deduction. Deduction associated with ~ 3syld . (Contributed by Jeff Hankins, 4-Aug-2009.) $)

Referencing norms: statement_comment/9bb1be8a11ffa410212a95e1e53398b69ace366dabd0a39c1e58a5a537fd1586

en · unofficial · sha256:555ba3e4ea0779197e3fc1f7cf4d93b675b233e5aef7077fb247cf1a9f4179af

$( The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) $)

Referencing norms: statement_comment/70ad60327aa493569b6cc72bdf6d7ddfd0d5664d5e2f48ef980115f8dea4767b

en · unofficial · sha256:4dd668f22bc521f7ce813bc264ddce5be2b086162219524a7e170c887e873f0c

$( Bound-variable hypothesis builder for the at-most-one quantifier. See ~ nfmod for a version without disjoint variable conditions but requiring ~ ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.) $)

Referencing norms: statement_comment/83951f1fc5a2af09868274dccf35adc5ba1997913e31d4dbaef65e2a787964ba

en · unofficial · sha256:846448add7c09ff64e854cbcbd4757f4556079b7ce1bc78a7282f457c2248586

sylsyld.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/34b272e8b96877bc4de44acf39b29c789d4b86e87d746c1096197d432f4bd34a

en · unofficial · sha256:3f6a41bcbb01f3219a7a72c9eba611ad1d60dd8017d6c6adc512db5d7800a726

equtr $p |- ( x = y -> ( y = z -> x = z ) ) $= ( weq wi ax7 equcoms ) BCDACDEBABACFG $.

Referencing norms: proof_reference/527d0480cbb532a77a2839595629dbca66dd10119ed3352f3b0da7c1f8b8ef49, proof_reference/86f7300d19818716298ff13d3ab727512fa8c0fc7d156a2d150aa0ec83909655, proof_reference/9aaa5c9fc45375ac7979321212da15ba461dfdf6d85cc64bb7f48cafbf9dc61e, proof_reference/c55d395f44e2278817208b41fef2c973e71bd4a50d7d5d126674b6066c1c7b2d, source_statement/fd2adce4a80642dcbdce9ec03fd2400766423ae004f041d0d990ba47d94beb35, statement_uses/0dd12515211c57b0c65128208de5c8de390905d59e11dae404669aa4638984f3, statement_uses/710ed6c093968b34c5dc24cd3314ad1cf0efd7f1e62f1b51a92dc3b9b4bc64aa, statement_uses/826978c1f6a600b11b91e3c18437d291013abaefad2222c2025b3ae7a27f7166, statement_uses/b60c50c0de5f84c8089dc208c61762ec5cc71eaa63bc109b951779388c007741, statement_uses/cf2a5050f39ebb592b7a649267868cabb221bcc171864543984268404355a5f7, statement_uses/cf710616e2b7dec0c6e0c768d8de70e037cbdb2bcc336ea0b1fbcc699d16faa9, statement_uses/d4b3c95417c16c5181ed4630a77eb80bfcd7f6255806c88802f62f9bdfa6ee9c

en · unofficial · sha256:527a1dc5540b563c7ebbe12c0020587ce73eb13bd48739a6c6c93b6d40dec7d8

mtbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/a333fe31dbbd0c6b2adc83347f1ce13f02894938ca3d47015db1ab9baad7486c

en · unofficial · sha256:e12e6df75fb7043a10568599319a3b744511f7732208def9f49e6ceb87df4d04

$( Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.) $)

Referencing norms: statement_comment/5af22d0225aa005b2331037c65b5cb13372901ad67bc37d02710e045c0dcf8b8

en · unofficial · sha256:8f4fd4c76b8cf69fcc39681d9fbc4b0e5f33970dc3523c42e16b9601853a4cf9

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Negated equality and membership =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/fd67a94f6d0e6dfd85b2f90fc2423a51ce22213fd1c99d548bca6f8db55a4582, section_title/ebb580b3087fdd3f58f9948a894c7740dc7fee93f3f7d55fa0562221e63ea94b

en · unofficial · sha256:01efd1732adf448860bebb325b680731e948e1731474928ae49ac2ba78b6b739

sylnibr.1 $e |- ( ph -> -. ps ) $.

Referencing norms: source_statement/debdc896153eb6cc923e4a76f7338355a352a2e37d03e5935d370c1193dca564

en · unofficial · sha256:2cdbb9f2a80dea8e68c3f8cf0b15c60abe2f08a56006d92c49335eb9039e8ccf

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Russell's Paradox =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/812b1f14f09fa74d1d74933dae243604b103b6b27ee51ca37f36372ae1ee16f3, section_title/be7724b6b15b4c9b0a2f06f32304cfdf45cc2d1b974bf9826f4af0034edabf5c

en · unofficial · sha256:463eaf3bad6168b652559d11f1d2574a995dde5f875697e3a40d50cad0461158

$( Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.) $)

Referencing norms: statement_comment/7a7b4ff1b9942bd3bd056865b6f20c29ccf965db7cf12c7340d8555d2697a666

en · unofficial · sha256:15e222cf19e7b943e3364c807eaea667470beba0acd09e33c16d32705c0fc27e

pm2.01d $p |- ( ph -> -. ps ) $= ( wn id pm2.61d1 ) ABBDZCGEF $.

Referencing norms: proof_reference/24c5f1b88672e54c3fd3524df027ce39d88bde80a10b2e2fb6e54bd824bb717d, proof_reference/9109c3366dbbce361867be1ae341124f284db7bcf4fcdbb5fe5d888e55d2e56e, proof_reference/96a93d1254e592a0aa71da9111ecd8af0a705b9efc854a4044d3888131821d97, source_statement/026d530b2236a5573205065bcd4a09979d290801ae101ee4c5c657f55e7f96ac, statement_hypothesis/7d129532eeccaaca54f3b99862cc811d4d57a26e2161d570a3089fc30d3b97fe, statement_uses/124ce1b03aeae9b968efddd72d589bab084f7b147498e5f32adffefc9f41468b, statement_uses/2fab005af03e8e8ab30df24f326f91e553cec08e62367a1a46898b1a09362c1c, statement_uses/6664d8c714ed7cf69d8f9d382838ec0c3d45f56521d22544196944a43eef450b, statement_uses/69d571555b6ff406d3117402dd530eb4e8a11cdb03808b5576f4ac5500ecf35b, statement_uses/7daaad9b457cd72f654ffe2945707349ec333ad8b090b88a60ee1a6b47a9a39d, statement_uses/e79081f295ea0282ef297e3b3272753e3b1b02f23f6d801c5643bbbaf1b2eca0

en · unofficial · sha256:6d37aeb302da136a7fc97902aa6eb3761a0df33ba3bc73541f75305718d167c6

sylan9bb $p |- ( ( ph /\ th ) -> ( ps <-> ta ) ) $= ( wa wb adantr adantl bitrd ) ADHBCEABCIDFJDCEIAGKL $.

Referencing norms: proof_reference/041e51b520feb2f160199026d20dff6e6686a730733e0b8a93db0fc7aff31eb3, proof_reference/1847325a07d02701600da350143f32eeaaa81c9f62089f3d4276b1cd1b0cb516, proof_reference/5e4602b2b8e9405b78a2e0969611680779d4f91aaaf27133461f43780b9cb8d8, proof_reference/e47f056279f29aead4aa0399d5799537ce2d4942ddcb3c5549d6849d50276d0e, proof_reference/f51997e4462afe9911aa342d5dcf49ac697c77d33410ba77eb1256357b71c495, source_statement/403904812b2d044d1d26cd268f6d93152a8a8d41b32892d1f1b0d2c2c23a4444, statement_hypothesis/b4f372a4b4dcff1be23cacea7531484767ea894be21d4c79cf073f4adce8954a, statement_hypothesis/ea5d39fb0124e8b1c70da3da1e6f3da6fe31f1ec71baa4dad5c52d3041d527cc, statement_uses/03703246ff6a4039ac578f458e1d0f1a782c674e2d897c5cdaaf211d193609fd, statement_uses/08bf17a78d73dbce5cd6712a7d8cb2d4a36ad7d2844fc68b90d53d9e86f170bb, statement_uses/2b185101f8adca2bb7b21cfd7ca1e6baf3dd113bb991a3fa7e19b207a3d13016, statement_uses/5ac28bdc124afc9e9493bb00a6031c148026b096f5346f0ab1786201a8991d8f, statement_uses/7ee6696c8f42e8aab67ca974e120d8106bc3735e20c58f177e0e49d7d1c6e33d, statement_uses/ab584c5b0fdd10f4d326636edf4da3fec0c7ca3123580d375ab12c8e410e1e47, statement_uses/bc9a4df48127278fb9e49d51859fae7a9cf63d940586ca9c549a48446bbbc15f, statement_uses/c44d98254b3699c84fe198493ba8e963dda682af621687b2e6ab9a6cde76e3ef, statement_uses/de06d17005a2ad1437943c33b19c891487fc00924db0962a83dc838970278f19

en · unofficial · sha256:4cb3f242d59b07a816c68cef0f8a48fee29ea9c66ae588f80f68948425b0fed8

$( A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)

Referencing norms: statement_comment/7daed3472b9e8b33a6474ac79848615b962eea3fa34e2e0dfd505d0e1ff832ee

en · unofficial · sha256:d1f41ff02bd0dd924d56b241eb99c85a720f90e7eddba299346f7ea3a277b5b4

nfimd.1 $e |- ( ph -> F/ x ps ) $.

Referencing norms: source_statement/ef1180595ffa7ee8da9866c8795e9ba72618cbd2a5c331377a797e9d4545ff5a

en · unofficial · sha256:ee5335e23560bc4bcff53e97a0f4e8ba21f4acdf963dd851c1d2f88f22822b1f

syl5ibrcom $p |- ( ph -> ( ch -> ps ) ) $= ( imbitrrid com12 ) CABABCDEFGH $.

Referencing norms: proof_reference/123c2ca125a289eb20abe8b1eb4457ab189b5f9cc3e11e3ebd655c319441bc5b, proof_reference/b0d6ecc03d258e81d60e645171a47fc4a15ef16ba39fc35549e249ffc0e23ed0, source_statement/fca7d9faaf0a8319bf5d654473d2f30469c20b1511a063fcb5771405c4585add, statement_hypothesis/10ec68eac6d79f0d02bcdaaf16c0ca2bb0e5c45cf01968f02de386a3f8391ea0, statement_hypothesis/e5ac71867ca76dd0578ae58a0c18d9d96d2b9000a82994cfb1aa8780c93ff497, statement_uses/04e7b36107306dbf1386779323618c6c2a65e4275215ff9b8fd43f5c58e83ab6, statement_uses/253b4b82842bb718ff6daccc04b84a979e760851dba062cd2373200f459fbdd8, statement_uses/7740d80b9be520320db59d42e9b7c173e86131983c3bb8c055aee8ac1dffb4db, statement_uses/b0cf7d8145b97a671d9843c969f219500c43a3cf0f9e323888a6dd66054ed367, statement_uses/b79deaf5c4cee16be32193d816aaafdb82f8190a4eded089879377c966e3e7ab, statement_uses/bb5aa2afb44cf952ac8ce359a411b6ced7a25dfbe7dfc7fa34c62c8872bf0109

en · unofficial · sha256:912d1ae2a93029405442daf1098bee32186e4d4eae82377fc275784818329792

$( Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.) $)

Referencing norms: statement_comment/09d756a75007634f90c58a0b8c4cf69a4bb8006962b6104c18096a2d3cf095b3

en · unofficial · sha256:34de94313e22d6cdc9f2c3277c866f0dd63c1f778be42d9c48f25f238534093d

$( Extend wff definition to include the not-free predicate. $)

Referencing norms: statement_comment/2459c303a7744ec05fb0a2eaa2603c349faea9487c503b9128df5fbf2dbbe69e

en · unofficial · sha256:d368ba6d352c1813478deb5ac418f87f963bc01a462ced8f75e06205ba01a05a

$( Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. The class ` _V ` can be described as the "class of all sets"; ~ vprc proves that ` _V ` is not itself a set in ZF. We will frequently use the expression ` A e. _V ` as a short way to say " ` A ` is a set", and ~ isset proves that this expression has the same meaning as ` E. x x = A ` . In well-founded set theories without urelements, like ZF, the class ` _V ` is equal to the von Neumann universe. However, the letter "V" does not stand for "von Neumann". The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the Latin word "Verum", referring to the true truth constant ` T ` . Peano's notation ` _V ` was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. The class constant ` _V ` is the first class constant introduced in this database. As a constant, as opposed to a variable, it cannot be substituted with anything, and in particular it is not part of any disjoint variable condition. For a general discussion of the theory of classes, see ~ mmset.html#class . See ~ dfv2 for an alternate definition. (Contributed by NM, 26-May-1993.) $)

Referencing norms: statement_comment/73e65e8ef78e8f2cd3c303464a324ccfbe3b53a24fd9de4ce5e55b5062a60ee1

en · unofficial · sha256:ffa42c767548b55593deb2ce0f3b7c41219a683a03234b3de3715614f545fbd1

imdi $p |- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $= ( wi ax-2 pm2.86 impbii ) ABCDDABDACDDABCEABCFG $.

Referencing norms: proof_reference/389b0212873573d2afde131079b666753236c3d98cab9e96924ea755ecd8fdfe, proof_reference/73b8fdad1657d6dfe4fd1d4ff154d1eb4b9fa432f795831f280287df1fa83b57, proof_reference/bfe28ec4ea8872b97dc37a8a67e7608d114729f2f3e0b208e16a53334efa16de, proof_reference/cef858a9e3408aafbbfe25ec8746a0a19bfe96977de93d2f04fc92f9b2428ef3, source_statement/4ff982e9ccbe2732938ce4b5c09f6c971121cf4aee99efa707ff909c53dacde2, statement_uses/02e4906f496ccb7982d8709312ea6355da0e4f9807f89b3ffb0fc1a2d951c4fb, statement_uses/16b028daad72208673e5c20142dbebc440bd5227f2bdee5cb2194b750ebdcb6a, statement_uses/21de7dd3a9fb2e1eaa9623a205b0a9d49b7a450ff4318331dbc24d5e1ceb2491, statement_uses/73c1ede8b267a1715e16a493fc10e073e643996a6ba1be331088c5b44e74100e, statement_uses/921ef323086d2d6019f5ea00ad6074f93e8d724468f2be9989dab35cdf6fc977, statement_uses/c4061721905a15666d88af16b4ba7e5921159aaad24ce7123c86b67bd7f905d7, statement_uses/f7f99a18cd02673c132c8ea42df4e00f9802302522b8b233840198757e66dbd1

en · unofficial · sha256:6b9d77becfd4cb28d6504824b43d5346fa157aa6badd8d458ff803bacef23ded

$( Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.) $)

Referencing norms: statement_comment/3e6a170be27d1caac4855b11125bba3d4b31206f0e13fc22ca6ae6a154ca2b30

en · unofficial · sha256:fc2e5247d20af100a6883d6e145daa1dcbcbaa9aecca92f673c6eec62755f3bd

$( Deduction adding nested antecedents. Deduction associated with ~ imim2 and ~ imim2i . (Contributed by NM, 10-Jan-1993.) $)

Referencing norms: statement_comment/26403582cb21198798b17275cd7f991ed055fc62ff28aa973237284d94a125a4

en · unofficial · sha256:61243b40f27ba06212ebfe142413906a68f84ab977901a90213e8286d5c2d70d

pm5.74d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.

Referencing norms: source_statement/2809c7654b7947784eccfb9efdb29d121a0df18c39b057aaecbe8f9738ad7173

en · unofficial · sha256:6b50dd38dd44f2e7586f27ab9f13a7f4fb74654ea0f719e2deac2f7ea98932ed

$( Conversion of implicit substitution to explicit substitution. Version of ~ sbie with a disjoint variable condition, not requiring ~ ax-13 . See ~ sbievw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ~ ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Jul-2025.) $)

Referencing norms: statement_comment/0ce9b0b24f1061f331c2fdae70d16a5455f5f93fa5a75baf07b4ed9fff951783

en · unofficial · sha256:b65c5207c5cfecfd88c4d96d86495981a89bab10fc6e8bdb823f404f120d0c19

idi.1 $e |- ph $.

Referencing norms: source_statement/6d09125a454ce6cfffc36ca81b919052ace681126810c28d1acdece080094517

en · unofficial · sha256:3feba525232796c3094113cbc48fbdd70b88c3ef32d6b0056fcc087f2a7a8180

embantd $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $= ( wi imim2d mpid ) ABCGBDEACDBFHI $.

Referencing norms: proof_reference/2d03bf77d2b97bcf1e258e5235066091b20f6959a495e65a28b1c81a105e987e, proof_reference/9388410f3c633a36c406ab1c4669a5b458dd20ace76abd06ce3d7d6d84e9bd96, proof_reference/b5fbda8bcb6e5a28cdaa11bd12971639bc68329f53c92b723e3e3b2daa19c0dd, source_statement/e471fc359cc2b1506cd2e1ff3da84e1a71c1828c106eabb5d48e8a8eb3a2e54f, statement_hypothesis/1e9cdc84b8029fb42602129590c51b81b542abccf873db3fade8ee145f28b33d, statement_hypothesis/df3107ba79969e72338d5422ea3832665b7a0b1b556621254999826103a937c2, statement_uses/21975b83509a76a5b1bd048940049411fc6e4a299b3ca4448a7eb27bfde38c4e, statement_uses/420dd429125389f02ba4a92cd689fb74399dc7c36ffab8a9dc4596dbdc24d988, statement_uses/535478a4ed2f30a7a9fc618006f5ccc05e3b8ae5a91a58c83d1f26c8b6598aea, statement_uses/b304004833cc27bcb79052d1f00a04a8d1e40d0cf1dc07a61f5f6f978ff80fc6, statement_uses/c272b473569301e71943ffeea2e3b73e51730a47ba28ff1686f586616550899b, statement_uses/c5fbfe6dfe2d3d3cc0e1d7a16b4624b0ed90025b127e3c79a37726d5ea356e97, statement_uses/ff1967c60444c4a9b9dec3b247f87be32ff121d57d43f4cca3fb1dbd9aa97571

en · unofficial · sha256:8bf5af048f6c6d0dc832674fef9ae3206da6da3d9d6ee8fbf0221042c97b3638

$( Closed form of ~ nfth . (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 3-Sep-2022.) $)

Referencing norms: statement_comment/26bd75bb470bda2aa8cb7f9cc40796e31ed86a716766cfe0c3661d368b3626e4

en · unofficial · sha256:c9cde7999b313e8458e97503a7f5ef99d66ee10ce7e4a5c407b0db9160490d75

$( A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.) $)

Referencing norms: statement_comment/02b35d4cab24e77d34d3e44dd61a8403ff1c133c5110eb984bf0c96f761f1ef3

en · unofficial · sha256:e0807aa1f3aacea788a59626deb0b7c32d06223584883362f8a7c56b06a058f9

syl56.2 $e |- ( ch -> ( ps -> th ) ) $.

Referencing norms: source_statement/7278730749a63c57d3653a563a10a87b8d8a474e6740af95aae5644d3bbefa2b

en · unofficial · sha256:b4871d687152525cb01bad90c304c683de97f090defe0c032184dc2b806b785e

orcd.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/59385cecd04dc9afcece302a962c7dd2d8e408b9869045405042db80ef30efb5

en · unofficial · sha256:61d601067c77b0c7f9a829f870f7735154476dc455c74c1fcbe6c52cf32f8a7d

mt3.2 $e |- ( -. ph -> ps ) $.

Referencing norms: source_statement/ba588b1bb857e873c31c041a93f9475b521c30bc232137434bfe45fc643ed3cc

en · unofficial · sha256:35e0c0557d7cb5b26bf4a048dc58c1e445d1918881fcb85eb592eabf16d2d33e

$( Weakened version of ~ ax-9 , with a disjoint variable condition on ` x , y ` . This should be the only proof referencing ~ ax-9 , and it should be referenced only by its two weakened versions ~ ax9v1 and ~ ax9v2 , from which ~ ax-9 is then rederived as ~ ax9 , which shows that either ~ ax9v or the conjunction of ~ ax9v1 and ~ ax9v2 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ~ ax9 instead. (New usage is discouraged.) $)

Referencing norms: statement_comment/477846b3d9a7a60a367e83c69c402a1005812474268684b151de58394823f0d4

en · unofficial · sha256:73f42c3283b0ba8635ef77e8cbf955177a6285fdcd1b9d64ee30c2beadb052d3

$( Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of ~ abbib , proved from fewer axioms, and hence is independently named. (Contributed by BJ and WL and SN, 20-Aug-2023.) $)

Referencing norms: statement_comment/70569cb2a9b0a0ae2f4a3883b82ef3a646df3a5528f09788aa2b2e01b9b4ba4b

en · unofficial · sha256:7cff3bbd52bc30872d08c9fe28573f535a5c0062104df39f51135e9359e2d1fa

axrep4v $p |- ( A. x E. z A. y ( ph -> y = z ) -> E. z A. y ( y e. z <-> E. x ( x e. w /\ ph ) ) ) $= ( wal weq wi wex wel wa wb ax-rep 19.3v imbi1i albii exbii anbi2i 3imtr3i bibi2i ) ADFZCDGZHZCFZDIZBFCDJZBEJZUAKZBIZLZCFZDIAUBHZCFZDIZBFUFUGAKZBIZL ZCFZDIAEDCBMUEUNBUDUMDUCULCUAAUBADNZOPQPUKURDUJUQCUIUPUFUHUOBUAAUGUSRQTPQ S $.

Referencing norms: proof_reference/038b7cdf1b6258ebbfdd4056e0f70430ee8e14824d6820bb0941d6474ff23df9, proof_reference/0585e672ffd43bab9b06d558e9de854566d32e5d24e277c3f75c9a202b8c07ab, proof_reference/21b6eb68020e1f932b61599b90aa3c16d0ebe5adad207c733b1bab7ddb48031b, proof_reference/5f273337dbfffe21af40a40d11657fd5262b8a74720c52f33a7447e99264d20d, proof_reference/6e693c3a70c7ac76fc2c076b1999551a6ad8e1edd1a19bde2a565a94147c27bf, proof_reference/79ed4944d470105dab37b06dae6d3e2cf822e579baf7d7042585c3c624ca383b, proof_reference/aa20229885367a611ad9f016493ab62c0db2656e24007aa1968f5bd9b5e83995, proof_reference/ac171e8bcf385a341e77168a06b475af6983a554c206d97e9c9bb02c72ee9eae, proof_reference/af75c61a2724b9a57fa092ec01c2d9e4bd76a6c8b255b0e400d11c1c5268abbd, proof_reference/d0e12501f6f0105c6222d086827700d58737e28f6040f34b64d9e08db1193f75, proof_reference/da8e5fdb196cd513f304f4094b42917ad49b5f8c41acff446b0fe7e4c0df4dfd, proof_reference/e40de124c511bd7a62e1a152194dc2f4433eb87a92d1024557566f691996b807, proof_reference/e9c3dff72aa1da3d78f345681f8af80c9e5605423e386942d25e4c0e6a6262c6, proof_reference/faddff0eb2d785ddabb3ae5a84fffed3a16f90981795360f3a2d352114476cab, proof_reference/fd5bfa25b6041658756c06358159fd3129b929d23177a0fb90b02adb00b18c15, source_statement/f153437633e5927daa0aab28114adcf6fc43be33eca989ff54c362ec50a2199e, statement_distinct/0978abbd7eb90ef72b8ed8d814abbb1c753de4c5d32c64e17c023c10b21d9679, statement_distinct/1ad18587ce7c751dd0d0e8229ef69f0446317cfa313f5aea4b642f6a9d704f9d, statement_distinct/49095e46fa4a676fd92c5d6140f6ff82b2a12e8eb517e9191d3107a925d59633, statement_distinct/7c9a8edf4a8b144c3d7f182a60e056ebf831a2012b8aac86b8d871b4f10924f2, statement_distinct/8e4a0ad93839ea1f6de483bf031ea2365c25bcceeba082989248d95b2a0c957f, statement_distinct/946d905ddf611cf553163b936d34660a7a849a5a2b2b2cf7660273edd92d9e0e, statement_distinct/cd6b150bfb461e657497c70882c7b6f64bda766e6e6df49ebd10c66121a1590d, statement_uses/0adbdbd1dc118ebde69987b783d3551e6a30845bec8067ad1b997588b0e74885, statement_uses/2354014cc9673147a3023de7c7ae05b1e8b00538b518f44e0334458132e42af9, statement_uses/25175e7a6decbbf4922d6d1b7cd12e7e34c8b8b01e22acfde57fa0e48f6eccab, statement_uses/3e968c5a5f64ed720afff0ab6b50700d46ed9d837cdae2ced6aebaca67e30f98, statement_uses/52d5204b07d2454f64fed19253989bda9b28dd3b228f5c39f12902c52a3d0ff9, statement_uses/645472f0e98b70750e2b5d07cae7530dff66a33c0eced2535b0dbbc97820d6a5, statement_uses/6599db17f58c230fc6fe230eb316aa23c85d0248d9214826ff64e366e8542366, statement_uses/6ade4048f7de74165438d1032091cc01a9cdedbc99dbf0c5aaee10e595b235ec, statement_uses/6d605d5be5d49731681839f13a8f8b700ce10b2c99d59464d9ca7bed9948bee9, statement_uses/76dc4135c14d8fcb471b79e0fe53923c03f9a484121b110a9ec96cad133f86f8, statement_uses/9010e098746e20fc39f194320ee1931c64b268e55d162d8b19d82e02f92da628, statement_uses/907a99f01fb6b4d5fb19e23d05b162dfd9f1d8282ef5f9645c34ea1c5a1b0f22, statement_uses/e8c894ee140e93743f1154384edc510b817386e0d61d1a0bb37f30da0ce09a02, statement_uses/fc179e58517f9cfe5dbd33daf9222e69f2cb61a3e68dd2604307347b1b4cead0

en · unofficial · sha256:ae0be7f0110d68b9b40901f7c12d3a44fd5d916745a79059cc03ca10539da8b9

imbi12i.1 $e |- ( ph <-> ps ) $.

Referencing norms: source_statement/4cfeac8a3295555abece04c1b7671eb2c3c35f61bc9398460d679ef636a6b58e

en · unofficial · sha256:98e6e3274de35e65f60e7c224a8c79ca9c5aaa0d8563425e3351387e29697f1c

vx.wal $f setvar x $.

Referencing norms: source_statement/db0ea9121624db3bb79ea6290df57ffce2094764edf4128ee090d1ce9a1cf5bf

en · unofficial · sha256:f04dfa2e502970bf47fd0b806d71998ba6065a3d807005eaf5feeef64e8e5c9a

pm2.61ii.3 $e |- ( ps -> ch ) $.

Referencing norms: source_statement/390b828c258dcb261bbbd6a6db64bfe5183950ad1d3c8d6dd2f4596746dc4be1

en · unofficial · sha256:6969a4823c42fabe06cfe513b054baf4b7d58137e33c431b239087099781ceb2

$( Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set ` y ` exists that includes the union of a given set ` x ` i.e. the collection of all members of the members of ` x ` . The variant ~ axun2 states that the union itself exists. A version with the standard abbreviation for union is ~ uniex2 . A version using class notation is ~ uniex . The union of a class ~ df-uni should not be confused with the union of two classes ~ df-un . Their relationship is shown in ~ unipr . (Contributed by NM, 23-Dec-1993.) $)

Referencing norms: statement_comment/a3b8ef9998500878fe973e2230cd3784c7a519e8ddf5701e5a3fb30abec14dc1

en · unofficial · sha256:1fa6a6f19a0ea7ec86c232d10d573eda7769bfbf90571c69843f6d6f06b764fd

$( The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)

Referencing norms: statement_comment/724b64615933dae03afe110113f78b3a12c2c65735b05b9c4a1537e2177bb924

en · unofficial · sha256:69d787a50dcffc9a6b8cb1f9b7dfd8a28ff4939d3d7f908b8e880b56216265a7

nf5i.1 $e |- ( ph -> A. x ph ) $.

Referencing norms: source_statement/0afa27bc124f7cac6967f03e1477327d651066fa7ebf386d8e8f7cbe11a393e4

en · unofficial · sha256:b87036023ea0fc37a0bd96609ce72abd16a19fec1fd1d62a79bf18c65774fb7b

3syl $p |- ( ph -> th ) $= ( syl ) ACDABCEFHGH $.

Referencing norms: proof_reference/f5407f455663f373673abd0ba0f4a1f0bbfbd07accd40c1d6eb69049f7a316fb, source_statement/052924484c79611d0d89722844996ddff94e26b8f084e4ddd4eb520bf026ffeb, statement_hypothesis/be4b93f9b3387378f896209d235d677448c421c64c591042e710d57320383cdc, statement_hypothesis/c8f3388fb8691189bee83707feb74f75d9f1c8fd2907529d2a1ec39db418b9e7, statement_hypothesis/d81112a022c9478e36cdb4e73583a0e54bf7790bb68e2d83e11b33a3a5939ddf, statement_uses/81df5e2952191adf5d1df916587dd99cd06ae9b97f4ab67284761873f41e1ee0, statement_uses/d1825e53f172fa03b8c81669eff316fd241a6db317547b1cca31e8b4f3a4c9fc, statement_uses/e141feeecd641664bcfefd1d59757d1e765d4c696d42936ad1afff720dc10b6f, statement_uses/eaef2993b37d74600ae4053f22243242fa9da0341b8105a42361f9d69acf3e8c, statement_uses/fc8c5f00803df1ad4e66cc6b6718b204cbec87c62ab307bebe982b4f74f5161b, symbol_name/848a2fe65bfedad3ad212cd55f03b61a4d4c30baf7fd1a613ace1c40e9a7c57d

en · unofficial · sha256:fac1cfd7399535f64dbca86e079d86a41a91791b364e1a8b0cc5d62284944799

syli.2 $e |- ( ch -> ( ph -> th ) ) $.

Referencing norms: source_statement/bbe2e9e91ccd33d24e9881f67b7215e3ff785b9781fd0c752ea934b7786058a4

en · unofficial · sha256:9df2756bbdf3ab55a81c606ada625c68a0b4518758344b54600ec70ad3e0dd4e

sp $p |- ( A. x ph -> ph ) $= ( wal wn wex alex 19.8a con1i sylbi ) ABCADZBEZDAABFAKJBGHI $.

Referencing norms: proof_reference/3bd535771dd35445a142d94aada67c768a700e256f3e2181b629400facd74a6b, proof_reference/50624e9f859c45b273df3d3624db5031e7aff90dbc6d2f23a5caf9cdcb903573, proof_reference/85cec00a29c1bfa6aea8513ab2465a21b14c9cb7698fa4ae011fb248f750dd02, proof_reference/9aef9d08da43eeec3709cb78cc475f5ace2d3e2623f51fdb724f4946b55dcfbc, proof_reference/9e0cbff2f4534c4af698d926f43585cddacbae7054a8a63e3223cb7327d9736e, proof_reference/be2167b34bff011b339fa9bd91dd690c73bd971cedaf890333142f9094e6e9c3, proof_reference/c92f4fac579645de032c6740348fcb0edb55c4acbd10be38ff2a6ac6860c55c5, source_statement/bd2f51012e503233ab50a2bdecf6c410bc845542eb007637766650b13ebff7a2, statement_uses/0d35a3a634f53a25b9cb4d4f5b7b98cd59c29f6014fdbfb75a43e52930b4dffd, statement_uses/1259e36856ab1cc302fcb3895a2e1298d342181b40c76c7f6ed07c11423c9f9c, statement_uses/986da9732821aaf4fe30f5b4d2f714fc884e41e2382db5f76a61e2b17e452ba5, statement_uses/a2b383bd73e54b3973b3d78bceaf4664ecfae834d9e819dcfd97fd33355dfad6, statement_uses/c7a490b1febdfe88d7290d94b850b856d940ea7c3e30d4095e04948eb3c63f62, statement_uses/e5f57dce91f025803b980e9ab1f962ae61b834eae033e63a53445bf875fca2c3

en · unofficial · sha256:4ea39d905f80204a8b7fc2bbd742c63c8e59a1ddcd99f88fecdaa533251c01f1

pm5.74ri $p |- ( ph -> ( ps <-> ch ) ) $= ( wb wi pm5.74 mpbir ) ABCEFABFACFEDABCGH $.

Referencing norms: proof_reference/14ee0db2800d9542a91a08908c97f27e701d50dbbdffd4f65bfc02cd7222da51, proof_reference/1d3cdda9a80a08eb57f376a4756bb1f2e881cbcdef8db002e40b0260e227730e, proof_reference/6167f4a09d4dfcdddb00fc0d0c58a189199c7cae07ecd91415fb5a19261ea55d, proof_reference/9a6837c531b887cf8881bae5d6430f1daf970248922d6c5a5b27c08f9c7d84b7, source_statement/5e9d40f55359aba4119efc5ce5936a68f2af509b6f6c2ac0ce5176cb5a5d8f24, statement_hypothesis/ffed8287d6fae63eb03f5a11e6775762b00224faa24cc9ea3dc08b6c5edf52d9, statement_uses/1a2f6bbe24284614a1be3a8ce381b2a89b81beedac4228170e463548e8509c9d, statement_uses/6b1aeee8f7f87637e6f03c651bba0df1a4b44abf39be57b2029e2b56eb6bc390, statement_uses/6da3bf90c620117f190732e8f441dd4a09a90c1319b6730d711a375808097c51, statement_uses/b31347c3c046cd0278625f66beb92cf06b578f9afad84006a340c6c7ba2cc28c, statement_uses/bb01093960f828c258b5e74f1a9c0bda20c646e947470af3546b3c936e7c031e, statement_uses/bb5251df78f8a46f10490567899720509c742063848e790837d3dc088423325c, statement_uses/e9377a4ce78fb9824b6baa338253991a23e27e41a973c6a01b2fbc9ffdc9489e

en · unofficial · sha256:463404f407c4568fc7b30aa20854b1a22a01ed3be3c8752f2fb0e9e634201093

$( Weakening of ~ ax-1 . As a consequence, its associated inference is an instance (where we allow extra hypotheses) of ~ ax-1 . Its commuted form is ~ 2a1 (but ~ ax1w does not require ~ ax-2 ). (Contributed by BJ, 11-Aug-2020.) $)

Referencing norms: statement_comment/d66cf8b153a375dc468faf44bb4fd6b68d530dd8dead1ba892ba2ff0702290a7

en · unofficial · sha256:42e1dcfbec8305e611016b22ca58476bc66efd10c4853643a075c0b55d870ba1

$( Extend wff definition to include atomic formulas with the membership predicate. This is read either " ` x ` is an element of ` y ` ", or " ` x ` is a member of ` y ` ", or " ` x ` belongs to ` y ` ", or " ` y ` contains ` x ` ". Note: The phrase " ` y ` includes ` x ` " means " ` x ` is a subset of ` y ` "; to use it also for ` x e. y ` , as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactic construction introduces a binary non-logical predicate symbol ` e. ` (stylized lowercase epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for ` e. ` apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. Instead of introducing ~ wel as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general ~ wcel . This lets us avoid overloading the ` e. ` connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically ~ wel is considered to be a primitive syntax, even though here it is artificially "derived" from ~ wcel . Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.) $)

Referencing norms: statement_comment/6a121e0dcc4bcb3f4d739a744bebffe12a0d52dc479a4fba4d0523f60e2a4e2e

en · unofficial · sha256:c6b482da109ab8c42009e6cc2cce90a2c34686d6487b2847ca1be0bc41df6c44

sylan2d $p |- ( ph -> ( ( th /\ ps ) -> ta ) ) $= ( ancomsd syland ) ABDEABCDEFADCEGHIH $.

Referencing norms: proof_reference/892da3b6b860b53eeb6964951cd0537bfe633c8fc8215ff6265b4152d58563f4, proof_reference/a4d95115dea9381e5687e7869c059c51d9da5d0b25475f3536c4772c7fd0f114, source_statement/b6998adca33bd86e8915eec944e15632170393bd574f8543537747b0567cbfdb, statement_hypothesis/84d96a324811dbe86c85382c6e06b2fe018459ed19f3cb2e9deb4bbc5b9def5a, statement_hypothesis/cafa1888c0f8ce5ff6a6f21e09a4f489bd197bab11c581457401b05782c98632, statement_uses/2d3f21259173f9756dde4bac323381b897a07ca297f162cdf3e669f9f0191fab, statement_uses/5726fe9d9883092d7839bbc2215fa90ffa28e99db7cc3fe1f502d952ba6fd0c0, statement_uses/8ad0e451bca7e60405512e4cde458c9b95eb351895798f147277c9c50926de64, statement_uses/955b34fb5dacfcd46dfc7c26f8fd2dafc8733be033e426dd7829df582d158759, statement_uses/ae7e236f68e49da0a44aea33e4fa3a81f2226a24728e430fda2564d644c2d483, statement_uses/cdcf3ae04f1be9c04c489676cae4d509dfdb222b5e8e161409e2e88945c4b33c, statement_uses/d1e316138f0b3c743723d8d9ac971a8934b9ca191007a0bbd7d7f8dd074b4bb0, statement_uses/f8216aed3fdb95dc9c6c7b9ae18a85cccbfa128ac95fafd7c6274da0208306b8

en · unofficial · sha256:cf8ae4836721f2b817ed31aab9ec8af58163731575a4d37b234a814fa313ec4b

$( An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) $)

Referencing norms: statement_comment/7a421bfacdee3ecaa1a817e670a3d4f23d32845487b1af2ee05bcdffbcb53cbb

en · unofficial · sha256:2d52aeec9c65c49f4205077a8642cf55977a399a2a6a5609cd31aa682a3a9a1b

sylibr.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/84b5008d891c0f69a13d730225ddee9b4ffc426508b1e5accab99944c82dee23

en · unofficial · sha256:78e49d15da7de744841e18d8681276ce13147aa11d05428655eb97767f9a3eaa

syland.2 $e |- ( ph -> ( ( ch /\ th ) -> ta ) ) $.

Referencing norms: source_statement/eeaad22707f3503a37c285e66e8c9eb86cd3c630a73c0d1bc73c92553e016b70

en · unofficial · sha256:4eaeaf4605bf54f74bde570e0f166795105be72774b237b685dd1fc32215ee55

$( Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set ` x ` , an infinite set ` y ` built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see ~ inf1 and ~ inf2 ). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as ~ zfinf2 and ~ omex and are based on the (nontrivial) proof of ~ inf3 . This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ~ ax-inf2 . Theorem ~ inf0 shows the reverse derivation of our axiom from a standard one. Theorem ~ inf5 shows a very short way to state this axiom. The standard version of Infinity ~ ax-inf2 requires this axiom along with Regularity ~ ax-reg for its derivation (as Theorem ~ axinf2 below). In order to more easily identify the normal uses of Regularity, we will usually reference ~ ax-inf2 instead of this one. The derivation of this axiom from ~ ax-inf2 is shown by Theorem ~ axinf . Proofs should normally use the standard version ~ ax-inf2 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.) $)

Referencing norms: statement_comment/6649ffb6277a255d1cacc114b0486fc3f57a73fac423c50e5c95d43b632feab5

en · unofficial · sha256:cadbf273bf90098c166168506e0b04ae81a7e15f643b353833390e54d81ae61c

exlimiv $p |- ( E. x ph -> ps ) $= ( wex eximi ax5e syl ) ACEBCEBABCDFBCGH $.

Referencing norms: proof_reference/721c3bb403cea21b8fb8fed9c3a4b8a54ca727100633b66c3cadb66242be2099, proof_reference/7a0b97e3015c75735c6f79830897fe34ca900668eeb3cf7fa2f9cbff9378d274, proof_reference/da6b24f6842123cf83b4a7d53534a8f2ddc445dde67ed3b16240fd9a5d92dc35, proof_reference/ed3901ed0c4521a395ff5ea3c2b742867a8292c6629b3b1d600666e90503248c, source_statement/87d7dc215b017d342030af5d1ad14d70a241e1cb28382db70e730977be47a8e7, statement_distinct/90757f653a835d283ac4e037c9556d3b84328080d3442e17c8a56ce6d4b22a98, statement_hypothesis/fb71845aedab18b2d9868b8b056c0ca103e6f8782dab7e0ff02bb384ef4aafcb, statement_uses/0f9ec25e841950b303e6a8ec58105582addcce58b849477bd7892bc24e57c0a3, statement_uses/3afc1ad6561a025b94e2ba05ce292d61f545ad2d9885185126b4e2a33658be38, statement_uses/77db92373412049e82c90bf911f81ee9c310775e8d507c7c2220db883a027afe, statement_uses/cc79ed569367a7a441fb8a1968a750631b6b3a6133a23f57bc179192d36324e1, statement_uses/cfe120a0987276001acaceede54b0628735880413e7bf5599bf143cfcbc60b38, statement_uses/e62c6f67197218e471ba96fe00ddedb127c6bee2b2e8fbc0c585b20891e0f9a4, statement_uses/f37f4f709c456026bed06ac2f1500f1c493ae3705e6deca08f61d8127ccc7d0e

en · unofficial · sha256:064c7b9bac08682fe62d7c45a736f51b192186bd6d0c3de1a2ce1802e217e712

mtoi.1 $e |- -. ch $.

Referencing norms: source_statement/ab0254f1d551cbb4221418253939e07915a57ccb56a205bdc3ceb34e9743989f

en · unofficial · sha256:4c4d892cb965dde7b26c99af35558891d0952bcc510e2100ccea80b96aef4b28

imbi1i.1 $e |- ( ph <-> ps ) $.

Referencing norms: source_statement/c4db73bb2df4ea96e0b3ba7792a6826b4a4118d0f553589bac77598a01279ca5

en · unofficial · sha256:84fbd29947eee4d3eaf957a38486c8506d0b6d0fe0a561c6f8bb354bd5901948

$( Version of ~ eqabb using implicit substitution, which requires fewer axioms. (Contributed by GG and AV, 18-Sep-2024.) $)

Referencing norms: statement_comment/f5ef6b70facdffe227e28181aa070cfab0502e59095e88eab087eaa169ee52dd

en · unofficial · sha256:099862836462fbaac5cf020bb16d13733c53cd7c3adf74a31414ce970fec7386

equeucl $p |- ( x = z -> ( y = z -> x = y ) ) $= ( weq equeuclr com12 ) BCDACDABDBACEF $.

Referencing norms: proof_reference/306c1628094c1d10657365a84f34c191114c73dde846902dcced05f2d1dcfd40, proof_reference/3c5176e33cfc88a9f3e7bef6c3bdd5769904828e34823238c23a32342de8f6db, proof_reference/3fca26ab70fe77af3154b7ebcd0e67e014a4e9ffcd845d12c24b51458118a945, source_statement/55f138813ef915dad0809f7c9f7cacd915523590f4273d5853ca1bd09020b4d6, statement_uses/13b66698d479681745adffe13b413fcdf89ca786a83cd4a2a9ea4b75d350a378, statement_uses/6e02e934963b312a3189ab2266cbda37e7d091a6bdd5c862ed3a67581b7c6a6b, statement_uses/72739afa8647305c17311336780c3956c075ba593534df729c7202a2c3073bb2, statement_uses/b598440b0b5a8c32c94e67059759b92e5b3973622a26af2566147f0174d00651, statement_uses/c2c2990676f2bb6ae4c2492b873143668f0c6da113b8f019a3f4281af6209a5a, statement_uses/c85a35798199dff1962b28f7accba2dd2440f54aec384600874dbb114a68bd0e, statement_uses/dc69b596d93c5155a0cce45a8225fb8be80f02c482e8ca0ee2a452815878d7d5

en · unofficial · sha256:9778e4eb1225108144d69e3e50d9d532ba60fbd5ea3bff274d233ed97bf6fcee

nfv $p |- F/ x ph $= ( ax5ea nfi ) ABABCD $.

Referencing norms: proof_reference/027f436680fd8135a9fb5497e598205f58bf4be727ec277efd8014cec27727e6, proof_reference/a1cab74746d9f2bbcbbd8a8c5ed913042da20e4ea7a1c43dac015d53694ac2ff, source_statement/7ed3a8b5929b189eb1b9c0ee830f4d7ecdb21aaff96cb8dd5286312f0a49b3bb, statement_distinct/695b905ad3a34ed0f3b5bae783dd3fe1e5ba4093f3655278eb89390a49da7f78, statement_uses/07569359b36bf8026bb9a75ccef15d16aa71361ebb868be660f7626f2ddf71f8, statement_uses/5d7c75e506f0812f58375d80755240cd4b48b1f9700d4ab97a58210ca5209d93, statement_uses/fb69b2263fd1a7c6d5b4c0cb731484bb78c2c7eedd33bfd96a13b26edd5e6da7

en · unofficial · sha256:105873effcc1450f88c457e2897b4a9ef33e6ceb3cc49c0757497ef86f9b7ef5

$( Closed form of ~ imbi12i . Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE__WITH *" command. (Contributed by Alan Sare, 18-Mar-2012.) $)

Referencing norms: statement_comment/1285a6be068758f0232fa2ac10b59af5afdd12c53c3446c99e1bfeeb917dc285

en · unofficial · sha256:9d13be082b0918f2a047b838d032b9c7d8026b7a44ea1f30a604715e4d2e5c0d

$( Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)

Referencing norms: statement_comment/6742002ea1d0b22a4ae8bccd9765e0468d54b40c5d7d19ffa924a112493cbed6

en · unofficial · sha256:985d89a8c86d76ff71add1d96b4c0d468361939f5c8d0e51c2a350987fb24ba9

$( Derive ~ ax6v (a weakened version of ~ ax-6 where ` x ` and ` y ` must be distinct), from Separation ~ ax-sep and Extensionality ~ ax-ext . See ~ ax6 for the derivation of ~ ax-6 from ~ ax6v . (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) $)

Referencing norms: statement_comment/6c43c80c03f8d298813e848b72d9725157f751c2dc750737604ce20cb04022bd

en · unofficial · sha256:b5df85dcaaf36d2225390605957ded936ff891710350e22f7503cfa541e59088

bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.

Referencing norms: source_statement/ca59ff70d75d654d602c594329bad00158fb59bda0c711443abeab9eba0ef920

en · unofficial · sha256:66d73af8fdf665cf7cee6b3c0463db5094a6843fc732ba3f49a7648c53433325

alrimi.1 $e |- F/ x ph $.

Referencing norms: source_statement/9990361c26406c8c85594272ad91525d6ed8c625141ab4a81b2af06dd35cb1cc

en · unofficial · sha256:d1473204bd4675ea646245ab8d2956b22136b6aeabadad2e9bbfaa4331757014

$( Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see ~ 19.38a and ~ 19.38b . (Contributed by NM, 12-Mar-1993.) Allow a shortening of ~ 19.21t . (Revised by Wolf Lammen, 2-Jan-2018.) $)

Referencing norms: statement_comment/cf4f5392f3160540bfbce5af63e514f027bc58d55b2a0e26b70fd334a6fd7b8f

en · unofficial · sha256:93fe20aab207b844babe3d63d78d18b46c85b1acf41aa546b0d69686e69f113c

wla $f wff la $.

Referencing norms: source_statement/d890a9c7a344868e97e1de6cf4cda3af5ee8a2a0c9b74a7a303989ddc3f67e72

en · unofficial · sha256:3545799e98f48068bfaaa28b5c159aafb7beae6d9d8f13d425005b488d29919c

syl6c.2 $e |- ( ph -> ( ps -> th ) ) $.

Referencing norms: source_statement/199b7bf206f6737bd76474e33c1418a15126347245d75692c2e18eaf0fbda502

en · unofficial · sha256:4530120d70c5a1a6476f5dca6d7baa983e7cb4953b841acf961a66638b83abed

$( Commutation of antecedents. Swap 3rd and 5th. Deduction associated with ~ com24 . Double deduction associated with ~ com13 . (Contributed by Jeff Hankins, 28-Jun-2009.) $)

Referencing norms: statement_comment/102bf1c6e1bd99c08df0ab2b3d49977c6aba4a3b596a2ebb948a4c1e9486b533

en · unofficial · sha256:32832594c2be80b7e35407f126484a88f850b0edbeb3cef2edaee8fcfd690592

$( Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ~ ax10w ) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that ` x ` is not free in ` -. A. x ph ` . (Contributed by NM, 21-May-2008.) Use its alias ~ hbn1 instead if you must use it. Any theorem in first-order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ~ ax-10 through ~ ax-13 , by invoking ~ ax10w through ~ ax13w . We encourage proving theorems *without* ~ ax-10 through ~ ax-13 and moving them up to the ~ ax-4 through ~ ax-9 section. (New usage is discouraged.) $)

Referencing norms: statement_comment/3d690c9f0415003bf4b90e51c80176fffac44c47e990a4a59c613c7b32944868

en · unofficial · sha256:bb8b5aa431d5fe9e471ffc7f89633fb655c1feedeeaa2c118ff276b114fe5991

$( Property of the biconditional connective. (Contributed by NM, 11-May-1999.) $)

Referencing norms: statement_comment/a2106253dad1f459ad4a8e41e3e27084c2503715d0f7ae7af13045f1a2908fa7

en · unofficial · sha256:aa22bb192bc4e57b2a231ff0a9d8cc525448de3731b854c10a6b361d528a0055

$( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# ZF Set Theory - add the Axiom of Regularity #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $)

Referencing norms: section_parent/d90557e39cb7f1d56a8aea1f44f0f785a3fdb06526f7391b298d5edfae41000d, section_title/931d82bd71429f78e816ab91971f148a78b7338110bc0407af09fbec88d1401a

en · unofficial · sha256:ab44687f93a0997c2f981f0f62a27963a3600359f03b494e5b8c7c06ad6004bd

$( Nested syllogism deduction. Deduction associated with ~ syld . Double deduction associated with ~ syl . (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.) $)

Referencing norms: statement_comment/9645a1d136926c0d013f36caf942113ddc44c422e9fdb4ea6ae2d9fd6217bc6e

en · unofficial · sha256:3e7bd116124cea80ecb7d6a29c9fd96ba459840544adef174181a51336a78458

$( Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) (Proof shortened by Wolf Lammen, 11-Apr-2024.) $)

Referencing norms: statement_comment/4eb3a21cbc569d596f2a9ee2d5d2ec35b3ed0151f51254d68953dcdc51cf379e

en · unofficial · sha256:0cc730913c858ddaa65706e26dc72499bc455f776007a61a1d7993a371d875ed

$( Two propositions are equivalent if they are both false. Closed form of ~ 2false . Equivalent to a ~ biimpr -like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) $)

Referencing norms: statement_comment/1ee93f267e9d98c0393b5421891d854f40c8ebe0cdd482fa689df967e3cfde97

en · unofficial · sha256:64193e7f6c9c2cc560c9b399080f85dfcc6135c938a0ab60798745a0e4c1c72e

pm2.86d $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $= ( wi ax-1 syl5 com23 ) ACBDCBCFABDFCBGEHI $.

Referencing norms: proof_reference/023e6c486cd9c4632fac29f34d715dc1b533c4ff686d7a1f4d5030c875fd35fa, proof_reference/0b2b95d8eb8df9b91b3eef5087101f38d39d3155b777c27abbbb22c3520b8026, proof_reference/1e743b723709f7739ac6c4c3fd1b73b4477b224151f14e18e0967b49d7c00a73, proof_reference/9cba84fb9812d5b53b50beb253418d4a1e7a4ca6082fc866f3e539803209fbc1, source_statement/fcbdba33617e9af08d800f8d484e90320f07dd029cd6cb84db788126df708e17, statement_hypothesis/8cc9e9140d50ee539fd5e3c81f5f988524b57ebc850bdefcf3146b196efda48f, statement_uses/12278317f554a40287c8106be2b708f09d7b957aacfc5a7b73084552f0b5b2ab, statement_uses/18aad841d947f994bfe5b3a55ecc9089adfe2825f975d8e9820dbb8d7d6d1691, statement_uses/362e18d2080d709ffffa86f32d1ae8cfeb080400efa81782ffa36447a9f3d3dd, statement_uses/38b0a6c3c5379847803cf8d93f0f7ff9f2148b880ea3837c31ca87d14a715fc6, statement_uses/7c3fd8faa50850396642be0fc9f0d403fe8331caa67bb564b6e628bafad625bd, statement_uses/8c6bb9b0dfe7fc7830069f7b19b8018a2c551ddb82f021877e7503de7272453b, statement_uses/bb98b535bd9cf6e7f86c1b37e215f8138d4f77d66a6916bcf8fcd8655e01094e

en · unofficial · sha256:475b61adf9ec19799962eebc1303795497369fef2e270f81cdcb4d178c425af0

syl2and.3 $e |- ( ph -> ( ( ch /\ ta ) -> et ) ) $.

Referencing norms: source_statement/820343ab8a5815156a7835d2ec87fd8724b80fd42efb0bc1792a6d636748f2ad

en · unofficial · sha256:07f9b19646b932ab5813a4d549a6cb0e463f5ed2ef9f76d00289615f43b0374d

bitr4d $p |- ( ph -> ( ps <-> th ) ) $= ( bicomd bitrd ) ABCDEADCFGH $.

Referencing norms: proof_reference/275533f1875bdecb0d26f6d175ce815dbb853765329b29836fe7d0a871bfc973, proof_reference/97ee571d42cd38c83ae122de555a81aae0e72b37d89ffadfc0bf2e773f949e44, source_statement/b28f1d5e4f766fff9b3ac427726409657f37feff984954c3f00c091946e66ccb, statement_hypothesis/1b95cbfa32a9b70a55ee48005fd84aa1285251ed435598acd8fe306eb003ae5c, statement_hypothesis/fe64f882e7bd350fce68d2fd913945c1c9160d1729d5c38f8f4c722d838c7f6f, statement_uses/34958e220765a9e2958d65500fad3acd488801eecd96ba63438037634dd9417c, statement_uses/57767d5a55bfcbc4183a81f3fae3577800838dbbabd0d85c73a3e6a1df6fda7f, statement_uses/58b256d2cb5d045a6bef277d5ae2038c48c3ec6fade269a2011043c8a625448e, statement_uses/efe8aa0dbd09e4fe4fd0b5f953052c8266658d6f0ed933c681af3bd7d5edae78, statement_uses/f23dd100016e4a554fae4a6c8ac82535c076c2c851ec756f33da2a2d79b8a130, statement_uses/f91ce9623185bbafe3525f641db555f91bdd3733e5339f465cabbd9964292f2c, statement_uses/fb5f4f8e0ef8cc8e1b815448fe55cc4f66754995c98a4f00f51690081a95e33b

en · unofficial · sha256:97f7117dc8d5380b755b8630b5f59c79c3834d7603ab8b1040b890fc9471ff77

$( Deduction form of the Clavius law ~ pm2.18 . (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Shorten ~ pm2.18 . (Revised by Wolf Lammen, 17-Nov-2023.) $)

Referencing norms: statement_comment/999d54395df7853729d0d5ed47f4fb68f206e15cbb9a3eeeb208508480eb9f61

en · unofficial · sha256:25122a79b412d9403d783158b14ccc5b90bdf099581897ca0041f3a0e4a4ecc9

ax7v2 $p |- ( x = y -> ( x = z -> y = z ) ) $= ( ax7v ) ABCD $.

Referencing norms: proof_reference/f959eafd5c88d94158fac184502f107a648111b006e7e1723d32b2096c240fdb, source_statement/22ac0bc49bb62aa0243e10a9e54123cb817b7c7f1e7e6a76195053ccdd81d23c, statement_distinct/6888b97d7bbb4902f7d67c7e838c8b2d12550ee2d9300adc8d9d43a635c5d455, statement_distinct/dff42121c8ffe4b2b67994623647d7067639ca2d184c884775add4a62434b5d8, statement_uses/1833cd5fcfaaaf979711d62a9b60a150b84d2d716b7d76851daf3dc8373b477f, statement_uses/347b6ed75b0efac6f4d8547b1d956667f7ff255c5ba5e3b7d97002a91a4a7318, statement_uses/56d6bfc22726df93f13f3b2317166eebd982cbe333752be8f029eaf43c536b12, statement_uses/5b730c3b428c339e21def3a9a11e1f7bef2b644acbd42dd5ecbab35d5b54f841, statement_uses/9d0dd62397f1a8d1c257dd30f9afa85bf367359379e3ff731d021c937c5f3f37, statement_uses/a2cc1b2d912cf995cdc16d41dedf8ac7e6dffaac76a546ee3f05c58a3f59f64c, statement_uses/a54a3447513ef9cd869cdcdc9c587c2c43c155b3aa70b3bfe1168a436aa14cdf

en · unofficial · sha256:1bfce46b73db9df0e37aca5238aeb44b41ffab4188b6b42bfa50ac5b31c106e2

$( Weak Clavius law. If a formula implies its negation, then it is false. A form of "reductio ad absurdum", which can be used in proofs by contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in minimal calculus, contrary to the Clavius law ~ pm2.18 . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.) $)

Referencing norms: statement_comment/03430a6798fb7818f41471c3af8885e1a25247441e6853611e01c84e0190a48b

en · unofficial · sha256:5f9c3b275519c019d9070a5da4b5942f24b2edde39f99e21cf08a244fb84297e

syl6com $p |- ( ps -> ( ph -> th ) ) $= ( syl6 com12 ) ABDABCDEFGH $.

Referencing norms: proof_reference/92ad89dc2476852395452293c4a834c8fb301e12ef72c71148ede853933e5f42, proof_reference/9307d5468eda7ba069371414ce26c94f5ae20b1e1b4c41aaaf94ec4a6422421f, source_statement/a92d991bdf0e983aa2ec7ef95186bdd02a3d32f234c19ae9a0ae687c39eb93a8, statement_hypothesis/a621fabf25822314623bdf096028b50a8518ed73f592de706af92c27471f82c9, statement_hypothesis/d6c7277577b0829e7f19b1eb3f8740642a234c2530f8500904b6c8cfc1607ff1, statement_uses/0df84df9a416a81d88d7790a5517b27fd9d0639ff756cde126faf7dee5ae9bbe, statement_uses/61fa7865958222d507edb5c57248cee3b2a84c1bcffe908c6351a78f82fc9b18, statement_uses/cec8efd393ba5544717674da9463dcd3655cde0e67d486ef9c0c6ff0f561dff6, statement_uses/dbdb6fd482127b5cc04a0f0c3ddcc91e15ba54d4a55463fc522596b291885983, statement_uses/df0825dfdeaf4da6e5e82a0a4e4922022a19a953598d12870af6fe2c37b48e86, statement_uses/fd56c564f1b35a57ce83ae2d61e77b9d63b02faddede5faa3738a672ed426e15

en · unofficial · sha256:2259492fabeeaeabec448e66279c3c044823c0407b3e5007681077788a274780

$( Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Jun-2022.) $)

Referencing norms: statement_comment/50144e005b0da5d82c397b4137e44e0d34cb4d6dbf6be232683672e2a5728db9

en · unofficial · sha256:b0040947dea2214f7917f7ee89457f7c839650b8dbb4c23c0a1595fc5d645f4b

syl5com.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/87c472d09af71ff9210ae26aa0be175c2947c189ef5c1e9ec33ada0ff870d271

en · unofficial · sha256:1cb9a9d519dd9a93b5ef3ddfb109f9be76d18aec3577842778ed594475bb6072

$( Syllogism deduction. Deduction associated with ~ syl . See ~ conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)

Referencing norms: statement_comment/e0c5d7f226b00b64d880c808a0758d9118d1caedc11b9494decbbc80810d2110

en · unofficial · sha256:eb4aadb3623bf58154331d15129d755d974f0d36064d27073ea4081f8cc0ea6d

$( A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.) $)

Referencing norms: statement_comment/dadfe1cc44c56a35f556f75e4310142049369dc1ea3952853e2ad6661a56d4f0

en · unofficial · sha256:1124ddfd32aa3d3e1d12cd8ad06a2f3735a5d1b921d0c1e712cffa36c550005a

con3d.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/fe2007ab0f07d773c29f7b800b7fef0aee07af679ef354d877d608faf5ed5837

en · unofficial · sha256:e6cdd0088ee7e8c206cb621efe67c9240846905c476924493fdd1cf49c0feb0a

$( Theorem 19.5 of [Margaris] p. 89. Use its weak version ~ alcomw when it allows to avoid dependence on ~ ax-11 . (Contributed by NM, 30-Jun-1993.) $)

Referencing norms: statement_comment/27fe2f8bb2b04e12a1975e9d5851f26205d1914d8e3de58a42a1bfe69146d8ca

en · unofficial · sha256:f8181a6eaea19eaaf2e93cbb261964e1c8fcda2f08152b29f4722655ddc5c490

pm5.74 $p |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) ) $= ( wb wi biimp imim3i biimpr impbid pm2.86d impbidd impbii ) ABCDZEZABEZACEZ DZNOPMBCABCFGMCBABCHGIQABCQABCOPFJQACBOPHJKL $.

Referencing norms: proof_reference/63ff3ce9e3daf3245bae6db5b0407ae2222ada301ccb8a381c02ff60bfed62a4, proof_reference/9771a839db3212b28c0fae3e7d2bda0d12a8662c39024aa773fc7e850e5f712c, proof_reference/a55adae1a7b21e806b2bd4f642291994f682aeed65c1ec13206e0d6acba80a27, proof_reference/c54f6d131b6c5cc5b24080d2766810a4586517a006ba41c4b2f09a4fac303d55, proof_reference/def9e2e96804e23a390d54c21103459a87f008a1d99cf697231090a99724fe1c, proof_reference/e2d0a0a63c8dc2fea4f53da3e68b22011270ed4464a0c7e565d5be6f6150dbe7, proof_reference/e34ff1d85232ee3d8f0dfd3fa2458770410c8dcd0ad078745d118f169b191037, proof_reference/e5ae0e71efebfc3c43c55fab8f1b8e6dd66a94dd24fa9fa15627707a3aca1409, proof_reference/e931645774e7c966b8516f9845f6322c97bdcaa080333913fe9985233766cace, source_statement/bb951e4aaf7662e77c6179902557d844dfb771e081d3a1b8d6f06f4b2809a561, statement_uses/160179c52124f83a78e57b5db0db24ecf6502b2b8411543dd7ff33753bf302bd, statement_uses/3df9edf8707b7206f38c43961c879792710a1701745af72ec10acd7ee5091f77, statement_uses/b0b56cd2acad4a32f2c6bd5d18063cf1731aaee42c645904d82f575afc1c1df0, statement_uses/bf9c6c701187449d16d81d80113a05c03271cdda854d6cafe45cc91cbb0786dd, statement_uses/c51f027020cbc5621996c5fb578708ab444b2f2f904d2a530ce90632d13e0ca8, statement_uses/c55603d36cec030959a874a5730f6f91ce4f83db64394fcd8fedce04203e6653, statement_uses/f5dd8ebcd7f23cdc0996c41ee14710a0709e7c340c695327e3f9d399a21de55a

en · unofficial · sha256:54bde36c940a925b0b9843932d58fdaafa877665343d0c408a0b163277d97ebb

pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $= ( wi id com12 ) ABCZABFDE $.

Referencing norms: proof_reference/062eafab15ecab6bbfc9894efaaced3e5f9f859d66e6b3c2ebd73aa1d27c8953, proof_reference/a99f5516c3fec5a29881e2024afa8cc9d716dc3ca8b3ac8cd468805f3faf83a9, proof_reference/f9bd214669d0592e1019329a5abd357361a765a6fb129d6cdcff7bd0049fd058, source_statement/90175b3e8e48fc4e797b805244ea2b433dee5d30bb016c2e5bc9654bb15b0151, statement_uses/25592df6ff446473564621a557bb2c86d28f1a7e127d6cd95b448aa3d2e67e60, statement_uses/868320e23b7cc8ee01969ba40df3cfb30b1f2c7734e022ddaa73b4a40f7f2adc, statement_uses/de4dcd7f938bd8628f3dee614b0c4cac59539e633af64ddc7c0dfde31aded332, statement_uses/e03a6449eccbaa6c8f7f0eba0449d25df4f5b3a3275f9e3181379d00a65fa661, statement_uses/fb1ba10590f796a6f69344044e53c6c5704d63718acbd0cfe0ab4abf34971bc9

en · unofficial · sha256:af91e7ac0edcb47235f86b9bd6d8e7b6330576c43c48984b95424ec21bc5929f

$( Deduction form of ~ pm2.24 . (Contributed by NM, 30-Jan-2006.) $)

Referencing norms: statement_comment/6cb1d1ca61ebddfeddd0cba53c735ab0b17b6ecfc7f30da7c78729ebc508d785

en · unofficial · sha256:9967b3e02a7aa323a91bc3f843c1d54f6c4e43c61d79e775be78ab996871faec

chvarvv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.

Referencing norms: source_statement/e942072a54fb069946c916d95fcd44f4cbe34df37578a229c563eca25cbbd8b5

en · unofficial · sha256:0b3c8e2d2dac72767c6706f1da148fb083ea185c4e6717b74974dfa1c927a087

mpbidi $p |- ( th -> ( ph -> ch ) ) $= ( biimpd sylcom ) DABCEABCFGH $.

Referencing norms: proof_reference/1586e7dbc27f480adbba8d2ac46f774740e494ee8da685773657133e47241432, proof_reference/4f91e7cf8418b69699791867363dcb9386739f6c315f3a89a8fc6e197e3c59c8, source_statement/0adce282bd7f12b0cce5b547c194ab10b4838088997f91bf13bb7777db19c8c8, statement_hypothesis/6ac612c473aea62c6e2ce3499c84882ebcb5bfc4443956dc325a3bb1cbe81b5c, statement_hypothesis/f397ab9a216e90889c01597436db1ffd53f30b48d11a563deaff27645f2bdc05, statement_uses/2d799da8901e578619523cfd62852eeb9c3e20d5f729e8e6b31361358fbfb03e, statement_uses/42c1975015597975e20a686394d4dbf35212ae88dbbd726a7d8aa661d7f6293d, statement_uses/5c6bb05f4d5a6011ca762fae65e61c5435bd0ffcf2439ea19c1d8f20332b5003, statement_uses/5f43ea2ea75e111fedd7e20f659ccb1391ab1e6284cf0075f171f8633f34f96d, statement_uses/7ea54eb1e7c353d4eb03233f4c2378822ff347e9b5dac09881530668d03a64ff, statement_uses/e7ddee812ea56da53180d218135489aef48dae5b4e7b4e82f956f23e6393de14

en · unofficial · sha256:6e856e89ff3e0b046f452d35a4cfe92dbbccca9e23c0d22cf043a9401457e672

mpan9 $p |- ( ( ph /\ ch ) -> th ) $= ( syl5 impcom ) CADABCDEFGH $.

Referencing norms: proof_reference/2726bfcff7a68aff63fccda848be442702f74ba0afef8e1aa60a4ff780885953, proof_reference/ca3d640bd72a927626a543684bfb69fe13aefb9279925ee5a8f44294887a69e4, source_statement/6f57d78b0b37672d25864205589ee34929946e49b1d030d81b267403c507ccc5, statement_hypothesis/2bcdf3fc0a9f646d803b9ebf94b2a884f7103f870f1294673559efafb9b5971c, statement_hypothesis/c2ee519bf1334fd8f7496752b6528eab637dbf9e054546ecf982a13d8fa876d7, statement_uses/17147ecffb2417730d6cfeb680308975e4e9e368d415c390232383d977b4681a, statement_uses/2509a47f95862dd7d14dfa1d576af4650abfae9506520a8a48627f7bf225b635, statement_uses/403c9aafdc3f6200d67aadcdd9249945ed78076f0e342101f8c9ad65fe9865eb, statement_uses/816b14bb52a5f71753cca8fb398812773992e4fa778717c3e7a8534ba31a0b1b, statement_uses/83cf98137be36127b7cbd141d510e4276e9a38188c823074af73be2a93d01b46, statement_uses/c46329418cf806a0132caea12310f5312e59bbe968966ee1bc7b9286836f2e94, statement_uses/f1e0d800a59ede66835c07a257507ebe7fc09c5e8bba76d1e1c3b53f4e2a0d89

en · unofficial · sha256:8cee45a8e796c68bf05ddd4a2e3e34914ec200c5f2a5ded5efe2f1f593c41a19

$( Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) $)

Referencing norms: statement_comment/a79d5a52aa218bd8aab2ffd98983d2c76ec319eca5f56cef553a9d019f09b74a

en · unofficial · sha256:4f0e80fd3c032df87cb8a01423b6d0c29ddf360b4fcca9c6856f52a07caeec3b

ax-13 $a |- ( -. x = y -> ( y = z -> A. x y = z ) ) $.

Referencing norms: source_statement/23828f0609953e03b29bf5549ae3ff56764f4f0f68764fde3919ee48306170b4, statement_axiom/9cfd61b878cdc67f7e52eb674a5c0e6b2e17c3a085288099aaaaaaf1307946f6, statement_uses/063213677ed576a579418778d1b3c9d1de0b9ce7b8162f70edbf451b529946eb, statement_uses/0834c438c5c22b2f759698cceb9861735283ca18054bcfd050706f6dd355194a, statement_uses/0f0a3fa638ced04e7d69f598ed098956393ea59532ec9e030b77190da1e30919, statement_uses/561f186320a65f0bdf0dc21d8dda0c8d42b0b64119101e6787974645705cfad5, statement_uses/56d522a49b75b425b41bf5afd6abc5f2caa6eb6d4b576c4bc01c95c9bcab03bc, statement_uses/6c37e7b7058bbc23fb80169a4abd84ba7738b473441a56e3b311e44fbcd95038, statement_uses/7c4d84bae60a79e704cc11f179d468393ad1ece825a43577e84b08acb3189fa0, statement_uses/8e76db6e8418f08841d9d5896d86cbf04ec37050e316ba4e491ada316b07ba95, statement_uses/cd7322e9ef5be2033c5e9c6d5ffd4f1ef0a23230315f7fc807944f293042e92b

en · unofficial · sha256:15184e8e2e026f6015f975521c743d6eb9127ebf683b906a052edffe4228df61

$( Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)

Referencing norms: statement_comment/6c8adebd5d55de93bddd054ec8f8b03efc8d8bed12028cf049cb6278fd84b776

en · unofficial · sha256:9d1f131c25229183cd0692485ec0d7f9cd9e427315ce3a745c24adb12f25a2a9

mpg.2 $e |- ph $.

Referencing norms: source_statement/9ea10d7fe8d10857bb5cdcd1bbc66f767b65a2b76f7be18f313f47197eb0b626

en · unofficial · sha256:2a94fdaadd884018c4e09c7f46eb777849be7e7b017e7c2da87d6d26bd4eb22d

$( Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)

Referencing norms: statement_comment/5889b7a0510c15c2939d08e413f3694101107d8329d8a0833d5d381da2928d0c

en · unofficial · sha256:3b47377621b67d1362631e7308b9977071cdee2c92559ab56dd6f80075e22ca9

wcel.cB $f class B $.

Referencing norms: source_statement/05bce1d29a2926aa1cb607b25b2084752e702c2ba919e18955dd078c192153b1

en · unofficial · sha256:b3d2460ffaa478a91d9c7142c0157cdf1752493e2f0f2ce34f8336c33fcf9fcf

xchbinxr.2 $e |- ( ch <-> ps ) $.

Referencing norms: source_statement/f2afce85940f74ef1308f25e629e3655a081c84a7a77482be1788591777edbd2

en · unofficial · sha256:1908e74bd7b610cf1c78db1f03835a9a820123b7a27ceac0c5e3ef3033bcfe91

df-eu $a |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $.

Referencing norms: source_statement/545ab5efe4c78782a6892a98892e1ecce618d506ae5ca71e35ac66d713fcc1ee, statement_uses/3b9308e7b0a49546ce0b5871bff6d00dbe4d6194ed00bcf0c80f18c0095e0085, statement_uses/658c7bb71e4f26f750244d05b8c4d880fd088cc10678dc202d0b579f56357d00, statement_uses/721ec7e5ab2e386a993c179b3abb696ae1ebe48f250b20ab294fc93e65f9ebba, statement_uses/73a8ae62c687682b352c54658cfa2551267eaf2c6f9b7ede7ca6ab6dafd990e4, statement_uses/7c2d9cf8c9579d9d05db1c316da002ddf2f38cc9711509536b291aab163916d2, statement_uses/c6ed999acffdc22bf91fe16cbd343ec8ebb1bed31022edd556f9d1c6ac16a8f9, statement_uses/efd06ec919d8c9347d7d80c642ea790d4cd0995388985661f5000948ecd6e515, statement_uses/fb4c6f442b6cf0e720fd64ecc27d3c4f8fcd5b23f615f1d5d58ed0f543b4ee2f, statement_uses/fe4123eeffefdf1ca29289c75d7a29b0237d7a270bf34ee9986a5f0bd09707cf

en · unofficial · sha256:d51fc7288834dc76a12ff4015c577242ae43a83878e0b08449ba28f778c6e6e1

$( Rule used to change bound variables, using implicit substitution. Version of ~ cbval with a disjoint variable condition, which does not require ~ ax-13 . See ~ cbvalvw for a version with two more disjoint variable conditions, requiring fewer axioms, and ~ cbvalv for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.) $)

Referencing norms: statement_comment/f98248b150a73d7d308d6c3f38fb5ecaa90e7982943ee4e215fe5066addafcd7

en · unofficial · sha256:f9748b8ca5e9ed1a10f804580b8582b38d6c471433275decf8cc72e46b9eccdd

excom $p |- ( E. x E. y ph <-> E. y E. x ph ) $= ( wn wal wex alcom notbii 2exnaln 3bitr4i ) ADZCEBEZDKBECEZDACFBFABFCFLMKBC GHABCIACBIJ $.

Referencing norms: proof_reference/1468b036bf1768f5c855d5cc5da286c1182c1642106ddd34ad6cf34400bcf0f3, proof_reference/2c6386e4da160ad6eee70e4db6adbaa277b5725e83678806aa54e8ab68c6c775, proof_reference/33980f7de42b6caf8347bd2f85d9720575d7f7d4af68f4ee756e518577aae6f6, proof_reference/83cd799ae8ea498723adbe68c0b9f6ea207ed8b610030414faa2af29aac9a6d5, proof_reference/9a2a582797ffca7964fe399635291710296096cf796f207c9b69f3f490c2d9d5, proof_reference/a1d28ccb9887d91322312f285121fb889b24fef4dc6db9f44159ad390b06c06c, proof_reference/d4f8cf18f315aac0692e7a7881814ce82270c31b7a57c2125330c2b53361ca8b, source_statement/923a635efb3f186a70138abf8cba087973ffedd3817ca290d6ccff8f37652335, statement_uses/231300f08680eb379c06783248a3c6c4e7ed6868537d7124324cdfcc5ad0043f, statement_uses/620fadd9645af580b78bad37dea976667f10cd91a3492114f6eb528250ac4222, statement_uses/95f16f3e6590c0bd02c8a3c5ef72664a84f206cdcb24c815c1997feba81ac866, statement_uses/b8957e37d595d871ba41d5480f5c871221abf87f4ad203adeced6fb7c240a524, statement_uses/bf4e3cfa0cba028ff50e1139042d7d7e76c3b44dfb0d734dc9532a1f1aca7538, statement_uses/c4dfd255ade400c415c5ab48363b2ae16ad37094aea76e0368021790cb90734b, statement_uses/f776a79847fbac3fc285945106b63d743343c477fe9eff63ea904b03ee976c9b

en · unofficial · sha256:96a43fd29a7641bb3e1ca48dddcf506cc589a16c5c2557400106a4cbb17d042a

sylc.3 $e |- ( ps -> ( ch -> th ) ) $.

Referencing norms: source_statement/be43ad35e33f4a7e76b910833dbd4a489c2c4bfa7ed13aa63a9888b597129cf9

en · unofficial · sha256:354ce5dae7d084646954a9b2022e143fb901f047ddcb12a00af6d8a8747702bb

bitr2d.2 $e |- ( ph -> ( ch <-> th ) ) $.

Referencing norms: source_statement/cbcd2745403b0e92e0045ed6235029aced3bcb38cfff5d0b6de27a9870b6ebeb

en · unofficial · sha256:489da293a55163c4aede3a7690a65d7c17f21e637ca5d48708a995a8f6896cd5

abbii $p |- { x | ph } = { x | ps } $= ( wb cab wceq abbi mpg ) ABEACFBCFGCABCHDI $.

Referencing norms: proof_reference/2bf7c409edc3766411f27710c78fee322faf5e33ad1537a3c46d9b34ff47b793, proof_reference/3015f5c40941e7227c0cc4c0ffd73dc4df42ba64a3e20a7898998d196a62f104, proof_reference/44966aad8ed9e4b12826f15c41c6dd6b8e4399639c10bbc415379e06e919501e, proof_reference/4ef7f47978b06a362c6e33fa736c7d2909cc246a0e37c4ecb784437873f86f4e, proof_reference/5252b45c2ddeba0417e4d92600f52b1c84810b1c97b55b9eea85a43752d99346, source_statement/386d235f9a2608b63ca1b5e02e77212553bb676b553bf71bf985a8f8e8fda272, statement_hypothesis/4a2d303278696e15165b63ec7bb0b92369327ee7df643827d471cd009609d849, statement_uses/2dceadb290d237e5534de9996a64279721558740c890ababf1c862afe19d587d, statement_uses/31d7f70b4c221548cd12d5bff174ad4cf07bc41f5a44c9122deadbd3267d509f, statement_uses/4139f0dfbbdbe46250267450aa5b36448d9eb2d03a4a8bcbbbbf2614e67b2eb5, statement_uses/8f3b463b45fb1de067451495d0d60c8e47d82fe63d31c61ea6a7e24cce016234, statement_uses/a34868fd8d029e00677f9ca98129b32ee0fbe6805d421ee8b070ba9fcffb4953, statement_uses/a9676d2507145e1e4e04877eeefb82bffc86cccb60db2b499eb9f22c9476868a, statement_uses/e61a4a477072f5a5dd25fd461c2f4008c346429596fc64a42812b2e23fc2be0b

en · unofficial · sha256:a2405ebfa0a07963bcb75a1ee5a1c1e0ee11fd506ec53856b1b61cf037d003c8

biimpri.1 $e |- ( ph <-> ps ) $.

Referencing norms: source_statement/c0dfe8ebc258ef6032e1d58c43ec85068e219b8f9bce1e9f45eeb4a19f01f95d

en · unofficial · sha256:96ec33a4d5a4c9ba05a92067c5a47f96e0623e7880da69c7467a0d1f418b1530

19.26 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) ) $= ( wa wal simpl alimi simpr jca id alanimi impbii ) ABDZCEZACEZBCEZDNOPMACAB FGMBCABHGIABMCMJKL $.

Referencing norms: proof_reference/293ccef62cba0cc34cb9528accf255f8606911ead264cf3e1b9b4d42c131d8c2, proof_reference/5e10a4bdabc9c5e9e9f9a0c27706775fce3ab1894eaa3806e16ace1818dc36c4, proof_reference/7e3eea0580474fb8ff59b1f608bca47c0c7524e64e09b3f2f5565349dabb9de7, proof_reference/9009a7dd1d958c3b7befa3fdb80fd8cb951bcad31875abe5cf48fdbd56bdd400, proof_reference/936e1f2bb23f140ab800e5b0849df259bda208c5c4c787414f1ab60e8b9ded0f, proof_reference/c3478a502698231259b4cbb7db4a819c0bcb2446e106a89a8fac30041e353892, proof_reference/c8321997bb1abc0036de935c36a9816ba425b567d56a31531f3bd57c2d94387f, proof_reference/ebeb0cfb22353fcd4b6ea7c8e7417bbefb362994f18cc15f6ea2a536f5e778cf, proof_reference/f212e36919d7d2fbc5d0740276b1873842c3e8ff0ac88df2058b31e6c0de37e3, source_statement/ab80161f021fdb8199929413cb068bc03c411fd54e35938a1553b257bbc56f5c, statement_uses/57f674e73201c05f7de6804643c420667d0c6851cfe40db0d4fb603190002b5f, statement_uses/6df64b5f44f6ed655cd67f551478413b645250bd0cbda004a965191be7c88eb2, statement_uses/7c3134ee2f7ce74a00576c6f9bce2627c54f04ac31fde0575cf42fd232893a26, statement_uses/7ca5962841683185a187e69ccfc30830c0080a7ea3ae244c1f918e2d7de02c4c, statement_uses/8292b85b7d83310c6beb8385a80857cb97bf7c9eeecd7d6e4c390c02b951a91d, statement_uses/9983649a9c76fa8e3c9769e93439448ff34e6ca921df32699cbb9ca628abe7d2, statement_uses/d3873aa322a259e626c98bc0df79e47e7fd7e069883acba45627d573ccba579f, statement_uses/ec1310199acbc20e90efe02a5b9cab29ea322bf974da20e1752564e5f51725fc

en · unofficial · sha256:de146196cf5aed123db23ed881c0eb45b5a5438f00f4ba3a22b0b39ef35f36d4

$( Deduction associated with ~ 19.2 . (Contributed by BJ, 12-May-2019.) $)

Referencing norms: statement_comment/a930efe266b96e3a376aa0d405706439c67748f041c01d565f2c3f258bb7858f

en · unofficial · sha256:0ddd3d8e293c450b7feb8988909af8e4afcf50705629dcf4c8a9d9d7e566d896

$( Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof shortened by Wolf Lammen, 6-Oct-2013.) $)

Referencing norms: statement_comment/851fee6aaee3ae554f5b1b90bd3f930968008c05d6cab3ae27fbbf249c5ef20a

en · unofficial · sha256:263b69ee0e9b4524771fd831c7d6ff40f9bfecdadb0d2573a6810b3545704d78

$( A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem ~ zfinf2 shows it converted to abbreviations. This axiom was derived as Theorem ~ axinf2 above, using our version of Infinity ~ ax-inf and the Axiom of Regularity ~ ax-reg . We will reference ~ ax-inf2 instead of ~ axinf2 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ~ ax-inf from ~ ax-inf2 is shown by Theorem ~ axinf . (Contributed by NM, 3-Nov-1996.) $)

Referencing norms: statement_comment/3d8a5bdc8269a1b4fab922dbe08e4dbca5eaa0b39b359abccfe5f151e352181b

en · unofficial · sha256:01d38822f735f3dffe65cf7f932d0f7cc3a924d3987e89fef6bf1e6d9f979310

mt2i $p |- ( ph -> -. ps ) $= ( a1i mt2d ) ABCCADFEG $.

Referencing norms: proof_reference/dd0fecc9228027d22e6e2078ce03416fb39ff2c4289fcb49ff7079627078b626, proof_reference/e3021a133ec214e0137932627f1174170aefb87db2cf3a7a5634a114b1d3a07f, source_statement/a6d9ecb4b69ec95b69bbf0ebacf4129480db0db45a87082e68c586d42023a680, statement_hypothesis/097b3d6ef3449725139deee9882c69b36ee33a58480268e7b41bbdbcceb6b4ec, statement_hypothesis/fc450fb2a238fa5ba5f488f25d539f818e4e3ecc3c33ee6b2c8a81f7c085fef1, statement_uses/039154548291f3f4753e28136ff98ecae70904ea48ea8b1c06c5ebe0fae769af, statement_uses/7bb1cf8f03b77fba559ea9a70baee051450c3c20b506c928ec55179c656c340c, statement_uses/8cce7357dfefca43f805961481c9bf39f518b2bd26cb5144d091f963c4ece5e5, statement_uses/960ec897763d6cc0b9d6fbd8a4a2fb886a35a3a77f44e92f2fbf9238315f1d1a, statement_uses/a7ef287c88070a8903222afff6da06178ad54ef18ed8bf7f13f83aedf051cda4, statement_uses/dd0b4eaf34cd2e6e02dfc913719491a957ce2b9063b8c94ff0fadd65713c3fc5

en · unofficial · sha256:7cb4d0160bb1fd88066dc9814cb8539c71bb3a66d18682323284a775e7e3b069

com15 $p |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) ) $= ( wi com5l com4r ) BCDEAFHABCDEFGIJ $.

Referencing norms: proof_reference/0f74d7d29825ef5e38e9e410d3df74f524691c5f445516b832911c15f9e6db35, proof_reference/31a55df0f972e3058c2c9e645dcc06f03b20d797ef3d720369ac216e8e371a3d, proof_reference/e6079023cd75da583e614a31fee29b69634d4b1296e4e538e37f074e755f4a40, source_statement/71ec2f279a3e92844882a1cbaba699ad56534ac2072366d48a62d44d4629ee55, statement_hypothesis/57219a128aaf0a5474f0914e8f4518eba4be3db2b7fd56b48be180a521840abb, statement_uses/1599e3976c8dfb114876459a68635629cb6634c697d0a15a08d138b41bd616d2, statement_uses/2e3c38a72dd0a619c5ea2b323aaf9cefa8b467b8bdb6ca27b64dab8299cfc21b, statement_uses/303cf59ef77157dd92363006587ddbbfb477ccd1b24135ed64e0add5c6edb3b6, statement_uses/35b753b1dfd280f5b583efc646c11361ecc0ed8423a9453a56eb9508fb3ffac9, statement_uses/56f7ba50658a724f2b5fb1a8017740b9f23ac679aec3ce755840463b66671a4e, statement_uses/6bbbe7414c233b3f50397a54fb54f34149d4abe43c2eadfa6cc5b3ba0a29759c, statement_uses/90fb63d073e966a480e7d2a6cb397480eac62d5e8304f8645bfb589ab48240fb, statement_uses/d9890a12edf006789763ac133312f59a3cc3e3bd06b5471e652933860325a70b, statement_uses/fb4e327650df03e29348ec92d4763b90cd69fd10033a493f6848c135b26d1174

en · unofficial · sha256:5b137562991e786134f630e6a4a2e1976ea46a2231e8ae37317094908b02503f

syl2an.2 $e |- ( ta -> ch ) $.

Referencing norms: source_statement/8e351408b36f892f8de7addaf349e1e58dd5a63e04b610b943a1363b4dcb089d

en · unofficial · sha256:3f12b45b1970a0517b64e8007957cd821804b17581f1b4bcfb0643e0a7329558

ax9 $p |- ( x = y -> ( z e. x -> z e. y ) ) $= ( vt weq wa wex wel wi equvinv ax9v2 equcoms ax9v1 sylan9 exlimiv sylbi ) ABEDAEZDBEZFZDGCAHZCBHZIZABDJSUBDQTCDHZRUATUCIADADCKLDBCMNOP $.

Referencing norms: proof_reference/0061f5064032be657097184d95e0bb01de6cf9aa852e36cd7576391a73d64a9e, proof_reference/055e06f56b3e88ae4834ca247eceb6f57f4e776e69e74ab0a7be4ab461d9444f, proof_reference/1a7232baa18fd51f980fcadf85149a2288663f232cd4caa08ec28ec73f09f391, proof_reference/1e3d6ebe4193b30c5166c7ca53e07bc14bb8e313512aa201aa5ee2bf108f8994, proof_reference/4b38ee0f1ac39a324d9fe41cfe721f1fbd27f4d90abe3d7c2ed7bfba34e3c20c, proof_reference/507382bcc65dcd4ff934e2504f5fc22460c7e9b1ec69ea856486807c15f28a1d, proof_reference/aae718ff089ea862df70037c93073c0a081b215379b53789ec8c2857eaf2cb7f, proof_reference/c66ba4b44ff39fa78ab66a515d38aa7b98e63d74a872a34757bf412cb748cd64, proof_reference/d2cac593e428c632e51e154c37eb5c520f55760f21171dea13809419962250ec, proof_reference/da3897d0a00ec230a694d434a383bc93852ff9d034525a589be487b28d56b738, proof_reference/ddbb4b57fb9aaec1b9795b0be44abd77cfda1fc4272766a5e16f8869c140d936, proof_reference/e13be631602ed7290a123b79edc9ef0819c6d8c57681ba6a2e5b44de5515e8c7, proof_reference/e2241981f44312a1bc0878dd94fa37bc5a2599f681b59cb53de0c1cd984a449c, source_statement/cbbba11f6790cdb232566755223f3e6f609229e8921478e648ae20da9d3a1e8c, statement_uses/10b820097b5a377ae39487d8cf531acf5d00441ab126336544a7dd791130039e, statement_uses/4f151dea8341e30d5c5313716172d085d39e96324a392b53a03b0f0610e64ddb, statement_uses/8344880de8bc3151d8c467fbfa2e5e82416c1ec32681603424ab96783e2956dc, statement_uses/b60aeaad515f8539711709ccb7b24787c36eed1ada9580e8fc887cb82d314abb, statement_uses/c7a652343394c9b1ec7a3844cfa0a1b0282357f8c389575c404edf184cfbc247, statement_uses/c9aaa20bc69df164d4ffc4d8136563b12def3af17333d8df2faf607c980fa897, statement_uses/ebdfb17b73d52d2e007c0689d0ff6b65a4974501728f75cd69e4b791c7893146, statement_uses/fbfaf23dba76187f4c76c8820625e257b7837d3e4944dd9bd5fb2062fdc88d9b

en · unofficial · sha256:bb8b5aa431d5fe9e471ffc7f89633fb655c1feedeeaa2c118ff276b114fe5991

$( Property of the biconditional connective. (Contributed by NM, 11-May-1999.) $)

Referencing norms: statement_comment/358db4e6cf7f81b6caf8f88df59d49efbe25f0604c1ecfbf37e9f9b9633169f1

en · unofficial · sha256:0ed3340cd833841c62c31eab6ff112ec8a453e282bea5a6a3eec3822cb1a0cab

mt2.2 $e |- ( ph -> -. ps ) $.

Referencing norms: source_statement/6855211fa93e8f841cdaa744a9aa12639e0f8f190fcbf9fcd83cab5700b6b324

en · unofficial · sha256:2e6085fdae66ef0e4cd74173980297d5e0d5065e35084aa7ee9b4a4834fe81e2

com25 $p |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) ) $= ( wi com24 com45 ) ADCEBFHADCBEFABCDEFHGIJI $.

Referencing norms: proof_reference/183fe8d9bb6b32c70d8bd8405c285f75f60595a2d26ac17ba658963178147650, proof_reference/49bacf4f6cb4500b61a7d62edd99a7dbe6a7f9a51bdb383f74b4ae39353915c4, proof_reference/7ec9603d213465ea5fd6110042df0728a6099fb3f09ccf46bca9a177a4bf24f2, source_statement/b36c8d6730383c5d10857c64062c966c3bfc84f2543949f15238d652309d9e64, statement_hypothesis/22da8059ff27a0fde327af71f24a64a77c3e2c3fa0c47807b0bb186f7f18fbc8, statement_uses/3716820a64216bb69953ac0638d9b4131ac59d9f1e7f3a81129a53fca3b89c63, statement_uses/38551c3d1f755ebed16e4e49a744cb3a8c753668385167a2d5276da1d5cfa5a0, statement_uses/4edb7e1257edd849e1879bd6ba75b12ecb909ba0a8b768f245ab18f3b4158efc, statement_uses/53a0e410ee9f0c1d759d59e18f87a47a679010c8d3636efd93fb93f12b567b66, statement_uses/63300e448326f4f30149af62a65ca105be0aee644efa3b40513090a1062ba1fa, statement_uses/681cc8e4532193c75260a7bcc92ab6688ee2270e150e38c0dbce02be8f5a548b, statement_uses/88bb414bf32c6716fe8cd74c5636a89e6345f88a49c001106856092bb6dd17de, statement_uses/eb9fc0954e2b76d1e653846a01d5f2ee6b00b0a4c91279a269c5314ab949e73c, statement_uses/f6fc0748e4315bb2e9322c19c191ad270e2d88d3f8b0682fc2b50c9cb182f8c9

en · unofficial · sha256:3dfcd6b08ec2067b867ac678755ca7233a40dc34a9099b22ac2edf3e7926d013

$( Implicit substitution of ` y ` for ` x ` into a theorem. Version of ~ chvarv with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by NM, 20-Apr-1994.) (Revised by BJ, 31-May-2019.) $)

Referencing norms: statement_comment/632c0297acaca14ee75cfc77902af510bb458f47c8d27b32af3a4e1ab2b2e940

en · unofficial · sha256:d2007758c0bf533fd1321c00e23a36e84ea5c0d3f95d2c17c51a88a97e4bc7c5

$( Relate the biconditional connective to primitive connectives. See ~ dfbi1ALT for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.) $)

Referencing norms: statement_comment/b4a40847963e186e18f03d6d6a184a930bcb478e9bb47255e1dda2fbc6dcec08

en · unofficial · sha256:2717f869288316016aeea5741d27b41507271c6ad74d92a42fed66ca7761e848

$( A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ~ ax-ext and ~ df-cleq . (Contributed by NM, 14-Nov-2008.) $)

Referencing norms: statement_comment/b8c31c19872cfce48e9396f753309a0e3a88f4676940281b78cbf77491fb74e5

en · unofficial · sha256:5b45ae07c70fbb9fef664db75ce1a48be25c466917d938455291e9730368c9bd

$( Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) $)

Referencing norms: statement_comment/754e3079fd2bbdad6790e026bf177adfa0ffe4598fa5ba5ff53387b73b4cfb54

en · unofficial · sha256:195fdc484a766869e0d851bc57b5bae791e5f8e376c5e1f980277c5a5c0e2427

cbvsbv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.

Referencing norms: source_statement/01a312c7bb060b9bb8f991fc96f637a483ee3b7ba645cda94cda1328be087b96

en · unofficial · sha256:b2e61d9c3d5f50b95d7ef9c846936c6bdf83450f21d58f6721272e31513a9454

$( Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of ~ syld . (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/6ff51d1bb2dd4f31ada9c0ffcd86bcc7163fdffa862662edb341a0a883c4d4c6

en · unofficial · sha256:34d8482e23499cf78df20381d539b3fbdf2739f4778d2874761a5a2d06255e2f

$( A double modus ponens deduction. Deduction associated with ~ mp2 . (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)

Referencing norms: statement_comment/26efc454dbe1b5ecc02335c0d2169bd39f86679246b4c86e4d7d92d69a7533ef

en · unofficial · sha256:174f11e0490b8ca7558b2e3972d766dd65d992e80493d40819061b8c55f186f3

$( Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)

Referencing norms: statement_comment/982e5de6336f2388e3bb7b0a4fc05e79a6be672c7249743d9cd724e018b5b1f0

en · unofficial · sha256:4d7bcd2495895add29f76335e770330f7b93a4c505ac117d5df4a29e7a6823cd

$( Version of ~ equsexv with a disjoint variable condition, and of ~ equsex with two disjoint variable conditions, which requires fewer axioms. See also the dual form ~ equsalvw . (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.) $)

Referencing norms: statement_comment/177cfcd7e5a49ed36473cafb22e142a587e05c85ad2e33ec73d7a69203df4250

en · unofficial · sha256:cfd1f54dc73a2a952c757a6e355136110280cde36a9d308e4af2469b82e0bbfc

$( A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) $)

Referencing norms: statement_comment/06a11ffa1dc8ee5bb96f014724b605e3283378e80b15e6b7c0bab4ea5b8789ca

en · unofficial · sha256:d7b2115b59e57b63e988ecc337f6d166bd3391864a3172510e20df16e7bac6cf

impi $p |- ( -. ( ph -> -. ps ) -> ch ) $= ( wn wi con3rr3 con1i ) CABEFABCDGH $.

Referencing norms: proof_reference/33b768e00db0566bd23050c55a80075b5a821e353a399f4837f2e6e037946fd7, proof_reference/44a07eff644d7d9adae9da5ba296f94c06f90087bdaaec97d7a58366044c855d, proof_reference/65dbff611aead70447f1660ac25ea54cc5061872603aef733803d3321155d5d2, proof_reference/870050193ef038b568a0e8a6c5ad355df27c08d1eb0b8cb88d260756b3498e82, source_statement/0956ab6da70427324851fca13ee71038e6bbbf2416c20c2b35d8aa2ac1ab968e, statement_hypothesis/a091c29961e17f3a5bfa766269a3407746d8a609f22a9d502ff0efc73ded4d12, statement_uses/2b11a75095c017ebbe78e1667aa4df52a4a1ffaf99ed66123461269ffad8aadb, statement_uses/39ec178b3e02fb037f8dbd4b6232f3982327e04959831adb249e3a4b169c825e, statement_uses/95a1bbf8375c560798103023a684edd1a6d9d02a3cc26e0298e64cccddc46f22, statement_uses/bfeee8715b9f7ba494d2d0289e15a8ab4a2c775bbb2db5fb3fd5c47c7cc8793b, statement_uses/c405f935aee60c699b9337beb582943d7cb5464cc8a5e828dd5757fc316bd7f6, statement_uses/c9fd85cb7b2a61aada00949774b9de08af43cfafebfffd61937f250ffa1dc8fe, statement_uses/e4795d51fb1acefb76836320018686172a7a97c2ae40f514e52ed4e0009879c1

en · unofficial · sha256:d9193552356504e64e325bda9bc3b17584e96b4b6701fde5fa53d8b9ee40d56f

$( Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) $)

Referencing norms: statement_comment/8e4221b62e1d527638761f3f2bb1649a208bf29ec290474984bd1ca493a98bca

en · unofficial · sha256:33dd8c966a90208b97c1c32737802131d658358a83130362b290a7e4ddb442a6

alrimiv $p |- ( ph -> A. x ps ) $= ( ax-5 alrimih ) ABCACEDF $.

Referencing norms: proof_reference/351cf0ab071002485039a42fd59cdb2848af199af4ccf28c6ac2dc2446e35b5d, proof_reference/5cc9a2ffd4db1255f59ab1b413dbba2b244737cabd83e42cfdb3baa317f3a7a1, source_statement/a545e6d58826e4bb5b317842e0b155e23d916c354bb697da97d7b2c1942b9a46, statement_distinct/e736eb2aafb3d2f6831be41f158d7483b2f3d41b8890b6c442f3b8af548cd54f, statement_hypothesis/f522b4a908093cb21d0642608b0421190d15d5f2136a4980a6e1b4d89d3be042, statement_uses/094355760036fafbb1a6845f1c2274f3e5a4e03997ce53413e163950807b9313, statement_uses/43e56927e0ab53ad022331ec12267d3823993d27be62c7afdef46561d52bf93d, statement_uses/6d071bd6de63585d0d899f5fb801241c1c019e65ef4c8b6c0e6e6e2ffbf7f1f5, statement_uses/70e872285bb734b7597d7e04f5e8e5d135c2ddf855bdd75b2a4161f76764199d, statement_uses/7fc12c1073698329faf2d555a6a269644e8d506ce4be372ceb44b85996ee47e4, statement_uses/ef204aaf7e394ff564e2eb37326001c9e98ba47f811d59f7d39edcd184732a8a, statement_uses/ffeaea46ef8f1f2a95386da6fa0e7bd6306cb8ab436cb6555ee9f0365f98c3b1

en · unofficial · sha256:a17e137da6788892a34e98d4a2cff1c0739fc14bb43af5088916bd053d8b4149

sylan9bb.2 $e |- ( th -> ( ch <-> ta ) ) $.

Referencing norms: source_statement/06003db425e037af97dfa6c2da10cbc5d88538b83f54db93404a7ac71e842385

en · unofficial · sha256:ec55d51318c830050012d1135a815e6a35f9274089c76005158880ea48faf605

$( Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)

Referencing norms: statement_comment/ac7e8402c58e59ad9458b67895de037af4e2d7f55784d4870fc18d0c7fa7beb9

en · unofficial · sha256:ad04f9a6e1c440d72ce824242e3c7903a4613ba5b70b8bc9f8a4c55b749d69de

$( Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of ~ zfreg ) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself ( ~ elirrv ). A stronger version that works for proper classes is proved as ~ zfregs . (Contributed by NM, 14-Aug-1993.) $)

Referencing norms: statement_comment/3d106ead39e77d085430277dea55c3e0fea9f323314f18f491ee9aa40cf9eb73

en · unofficial · sha256:a1b1b1281c37f270055ce8fea9a741b6d36173ba8ab9aba309acc2e1bfff2b08

$( Inference (Rule C) associated with ~ exlimiv . (Contributed by BJ, 19-Dec-2020.) $)

Referencing norms: statement_comment/98c9c431ebefa6860c90ab41a9a5d8df79aa97323bb4a3e5421322353386d165

en · unofficial · sha256:2da1aac66c9ae46bcb6bb85de50bb8edcfce1795de039912978a4a8891564944

$( The Null Set Axiom of ZF set theory. It was derived as ~ axnul above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.) $)

Referencing norms: statement_comment/9eaaf2f2d20e46cb155f7d3a899e588515625e50d80c7dbab978c3fe6ac44359

en · unofficial · sha256:25db5eb07f3f5df75e853eba1f310e1c37d0f1bc64c779a056d23bff646039b0

mtbir.1 $e |- -. ps $.

Referencing norms: source_statement/63745ad71d83abaf1fa4f03dbb6b6a074f415031286117d57bbf2fb0a351a3b8

en · unofficial · sha256:1aae46cf35e10b1bfbaa3b40cb6e7e59361c057774ab9b6b04a72921f19fdde9

19.36.1 $e |- F/ x ps $.

Referencing norms: source_statement/4b78c0783582654e1da24db2bb1c13a1470f0660f62d0f7d63c4431f6c61bdce

en · unofficial · sha256:a23013a69015b98405c225b8b8fcd9a73d51c5db3fb49c0163ef15e69094131e

mobi $p |- ( A. x ( ph <-> ps ) -> ( E* x ph <-> E* x ps ) ) $= ( wb wal wi wa wmo albiim moim impbid21d imp sylbi ) ABDCEABFCEZBAFCEZGACHZ BCHZDZABCINORNOPQBACJABCJKLM $.

Referencing norms: proof_reference/098e0bcaa986546cccf4ac22cfd45a3b62ef8e26d3b04307f5c3e28d21fa3d4b, proof_reference/5b1b7d2225a1e30a632eee95b6ef15e8e34f71f3617d5976ead92b165189ffa1, proof_reference/6dfbffc49ae4b23f23b43706e456f42d785326ada68f85c6b6989da4a8ab3120, proof_reference/73e8c4453e5665c61285ea1f2ca3142d73f23fd393b53b1f1c2edf45f425ba85, proof_reference/7ff03d65916dcd1553386f3e3c58e45c3a0cb75383f748098d81962df91b04dc, proof_reference/82cb5f6f27b6c281fe5ebc43de9507950969251d9ef4646d06238475e56de71a, proof_reference/8591da2f303a3707025c8496da2d213e7e3ef8043aa00d3acca5a5250285e6d5, proof_reference/b138989948da8d3fc74ddcfb27a88c265fd516ae4bb07658c94bc74d275d0556, proof_reference/c3f9fd250d5d66974bdcb866275bbb39ab5f7a758f6f18901a1cc91a1d64885d, proof_reference/f5238869d106a4b4f2b80c41c3565bcf0244642feca309a220614412dc934b2a, source_statement/bce13d9169496c2fade4e2f0e3ec7b38291805b2b709aeee2b4c72908893627a, statement_uses/1d0067a5fb560154dde85bfca3d58470a4a7b75ddcab266bc270dd538a89a438, statement_uses/4c52d753847e5bb68b3ca132257022bf6f5a9136ec5451c5d8c678496260595b, statement_uses/50a03b1c7ef7aeb28cee50146cb3bc4dc55f44082574bf478633781b8b04de7c, statement_uses/5c5d6b62f790d0405868088483d6de97464f4e076565dba52cea9212a19bf37d, statement_uses/6015b162c65b44c71b8e92fd7608a493cc85f29657dde4a03f0f5c9762367845, statement_uses/6761e31eec31896e1391de84dd506880387e2bfd55157ed7bbfd738c231289c8, statement_uses/b909a1cfba181a93bf248f2aade23ff3828bf09db5972c9a83cd571795e264ff, statement_uses/dbc08e82561c20c1946a3299ec4a0383e648f3b1ff279d10491750ccabd5d641, statement_uses/e1a97986f4ae8f65e5e6cd6e0592f201dc5ab8479e245b700b9a7c6687dc7a2b

en · unofficial · sha256:281825bbcf26b83559b56ca5706440f2696cb9d6bc9bf25e52a99db45fe75adb

mt2d $p |- ( ph -> -. ps ) $= ( wn con2d mpd ) ACBFDABCEGH $.

Referencing norms: proof_reference/0b651b40a9a86687f4f3cd3940185506a9f18065e196a6366919b55275c2f9b6, proof_reference/8769626c1b6508c07bf1db539ddf1a376c801fe31caa84afb65642a667166c3e, proof_reference/eba20a6315e6a3f892cc8b41dfbf09bfaa4e45c48291caac4300469c7510a4c8, source_statement/c047ddb26021bfb27d0da7f663f4094361ead6a6b0982a33a085016e2382a7bb, statement_hypothesis/a2ba000e6313a2c0c58bb9c60a411a392bee0e53f1dd9f058a30aa65023a0ecc, statement_hypothesis/ba2cdc4ab607a4d51b4bef9a756c30e87f41187c868c48f411948d364a3d70d3, statement_uses/4472394298e181c924313d57d867846d3dfb4b01fde48da7333939fd1cc1925e, statement_uses/530f47327dca683ea1f7fe7deda475a96e91ca5e39e3be4a4a521a32d3ba0da4, statement_uses/5f8c5978b45c7ba829fb94ade2c6b3602f63f3855be692edf88d8fa025aec5e7, statement_uses/b09906d63e02e593c90862f33e4b0d1467a869779a63c8e72c942a126d3a3c5e, statement_uses/cabb3666f352e97ea473a1c2e9e277459f434090e40cd13bd57e46bde31e147b, statement_uses/cb96e6b83cd5217efa33967f08e915a3474160825fa5a78dc69468aabbfa16a3

en · unofficial · sha256:68b2e5b44b885cd4ff721f43de1215bbd26037f8ce55126b882e337efa890fed

neleq12d.2 $e |- ( ph -> C = D ) $.

Referencing norms: source_statement/1629d2e3c4bb1c31c5333b45a12cc4fc2efa934cbd1470a0d2629714baf1ee6b

en · unofficial · sha256:a54098a3b475b1bea74949fe49249dae7e322d5a878d9d3907bb40312b22ce8a

elab2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.

Referencing norms: source_statement/4d77d7971ed478cc04d4db0e72884bc055a859f7f7f9cde0d6a4327566742420

en · unofficial · sha256:403836985ca6fbf1beefb53092799f0f0c129fdf162527cf57a9fcb3091b5fdb

$( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# ZF Set Theory - add the Axiom of Power Sets #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $)

Referencing norms: section_parent/4a6c78de35e79375aba967218c3ee182ad0953d5adcdf44204a92553d9f3168a, section_title/800c0c9f04800336cb2ee5163836b28f049a9adc9c745b36f60caaa3bb9364f9

en · unofficial · sha256:ab8f053063c1d9a929d5d29db189e2a5001999970d01daede0838b0d005476c0

$( Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.) $)

Referencing norms: statement_comment/e00b8c9b7cc7efaf82d221c6a29e532ddd78eafcc1552dd9a8315126242ca699

en · unofficial · sha256:915401791e23489af503a2cad9d6663a94e4729243e0f6afd1bf36b7b2c40fb6

$( Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) $)

Referencing norms: statement_comment/26e17d4076102ceae501c7f954d5f9d5744a2044f263759d380aee8ed8873c4d

en · unofficial · sha256:75363e53faa8b59b74a2e9afb4f1f0a38415769dbeaa5871d327c8702767144d

syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/a92cd4dab9d9e0af9a30eae100763f00c74d607ac8d64124db9ec7bbfbb82e32

en · unofficial · sha256:cabf084f17bee0a3053d21da4932b8806af9a023197d71cdd461cdfa22f096ae

$( A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ~ ax-10 , ~ ax-13 . (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable ( ` z ` ) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.) $)

Referencing norms: statement_comment/03f62927094a97c0db9664881d71b324f5f4dd6eaa1e9c73dfd5322c36d40f74

en · unofficial · sha256:c5c03ea5b728378cf08d86113c67199764cb22e0e23882d8fca60624db6059fd

syl5com $p |- ( ph -> ( ch -> th ) ) $= ( a1d sylcom ) ACBDABCEGFH $.

Referencing norms: proof_reference/82deda0666272930bd7f8ab6c7c551828ae4d8b1ee8cb69b43c43380213c331e, proof_reference/b9b331aac23afa755c419f69ff86694c74f5debd8cafe9da561164ee9da29dd6, source_statement/4f9765928f7ebf6eedc3fee9d4e8f2a054b3ae4b48dddc92bfa2bb03e2fbfda1, statement_hypothesis/32bffebee038045fee649383292dc06af0e7ecce493c58d5ea241507ef2eff76, statement_hypothesis/7b92a79f4452b12cd638a1937b3f74783f685f08640d7e214fa6b10540836079, statement_uses/20460e0e5f0d02c61d3d1b7e73d1aefcf8b030b87c8d2d694cad3368f320b3e0, statement_uses/278c6b62572ed51524edfaa6f842d4dc9851eb73a0f139af24ebc529eeaa7350, statement_uses/515f9aaec48d59bfaf87296be04e48c139e9800913c3df9b902e989cee1e83c3, statement_uses/8eb6c64a58d62e5de3e881679b911f6746087f2f675a772135196fe0c014cbed, statement_uses/f76e22fabbc53c4e597ff38e4f38108bdd831ca1e0457fff67544f4b5d428996, statement_uses/faedfce930bd51174fc06fb775a7814fbd37f639509ed7da2800b88c3c627eab

en · unofficial · sha256:b32c0e0633027564526e404ec91adb158caac68caaa7ac4eaf222ca6353cef5d

weq $p wff x = y $= ( cv wceq ) ACBCD $.

Referencing norms: proof_reference/16cf30719909852ed87cd5ad40f06d0b3e7394ac45a22875dd2b56e139c734bb, proof_reference/c05b4c106a975709cfb63d64016e9c13382a91884dc5a295c41ed526680a9361, source_statement/029e3aff65fb3d33ecb789ef601204d558e1f56dffde827945ec59a72ca0deef, statement_uses/903f2243367d0bef9740b25b260b7dc6fb2581253ea7ebc5def28c8e38a34b44, statement_uses/f2ce3e4922ed1751fbc8ab82c13d09d1b2e9d32e8b3e74cbcede7e6cc04dff5b, statement_uses/f46404631ac03e9133a0ab055577d86ca5c1e7a37aa7df12033f657bd0bcfb74

en · unofficial · sha256:a7a263ca55ed61dbab6d8698444f95aedfff7f926a4fe310a54dc2e2ea82792e

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Universal quantifier (continued); define "exists" and "not free" =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The universal quantifier was introduced above in ~ wal for use by ~ df-tru . See the comments in that section. In this section, we continue with the first "real" use of it. $)

Referencing norms: section_parent/149cb371d63f40fd680c48af85ec9b6d8432abeeaf834958126b3412507885f5, section_title/c644e5724d48a6683f3559df010cdef81c58d3f70e65e67e93c7800eb67587dd

en · unofficial · sha256:afe136103b208cfc7a9c43b166128fea39dff1fbee95570538e963933653b64e

mp2b.3 $e |- ( ps -> ch ) $.

Referencing norms: source_statement/59106af7c1d952a08656c38b4069d58d917764b08ac76f141820230b062481e1

en · unofficial · sha256:6f48139472ef9687bfe380acf85e2db150756a68703d8a8da181beea45e3bd51

impbi $p |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) $= ( wi wb wn df-bi simprim ax-mp expi ) ABCZBACZABDZLJKECEZCZMLCZECEOABFNOGHI $.

Referencing norms: proof_reference/14dff3551ede6e9e8c0d5e0df5b1757582c70122160977fb43beb9ac8d2615ab, proof_reference/20e6f018c57ec7ff8c5151ad663a91dc0fb80bf83584cc825c3c51a58b248373, proof_reference/24a4179c2de258e89407a863f661d2898ef12a8a163a1891741cc96b9b02518d, proof_reference/668ac675aa45e21858a15b356942193897841eb1b242934d8580970efac33b0c, proof_reference/8b0e6cc8522bca43e09fcc7dbc09afbed35a35bf6593b8ff42ebbb189e0525e0, proof_reference/9d7118846c7d509b8a36cda5a93432a366671794925322946bf4bb1f1d72cd17, proof_reference/a29dec95c2f6654db857feb6c4ef434a44c2fed49357e6ee3fe9b059b87d8939, source_statement/9286efa76cd620a387e77a443e851833b9cedd9d7cd52453c7e9f450df30e546, statement_uses/011da9ec9afb5f8c2904475160160c84c02d0e26294e34a50b4b8b2367878619, statement_uses/353fe0d4219b8d0d95c543bd829defde163be5c600fe202b35b0fff83d4dcf79, statement_uses/54d8e6047ed513c64b5d733b625890af6eff605d9b9544c2ab5100f46f928b6f, statement_uses/9f6099c05462edfc7c04c6a9345dd73fdab505d1df2d96cb6b39f3dd7de8cd93, statement_uses/b3e84260270f2659127b5580a866859da55280bc6431ecf5431d2575b79d57b9, statement_uses/c375e0c70ea8051b03adc6c80738eab477a705992fa5136b5d0155328a7abd9a

en · unofficial · sha256:8f4a42d3709e2c3a673fd9b638879ed883dfd7c7bce06d3edc6ac1c32830faae

alimdh.2 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/d4b3e064b0ed9428a3620a092983749827b68c71a865b1d3c0e750e156a45aec

en · unofficial · sha256:3847b97e74b48543f3526ebb3adaa5ebf7d31b400afae88c03b37b64128fbecb

bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.

Referencing norms: source_statement/20a3e43ae77892d65310c5d14844dbada1a89f5c7f9021fc16f08523ece3c28f

en · unofficial · sha256:cb723a95d5397683eb8a5404b4328971fbc52863aa088800976bc6f5c69776f3

mt3.1 $e |- -. ps $.

Referencing norms: source_statement/ea74f3a0c78612e842312d0a33ab42380996929f5dae9086dd44a020dfd645c4

en · unofficial · sha256:fa6abf18eab97322c4f48bf9fdbbbd5b97d417d23cd508e3cf78ddd83e14e995

mt4 $p |- ps $= ( con4i ax-mp ) ABCBADEF $.

Referencing norms: proof_reference/9408433c4c26c56d2cd9f6d6e1430b8862d93c2fb349401ded505322cb200f4a, proof_reference/d51687cde6c257eb9f6b09be26a8e89be5d39beb2fcd3ab8f5319458d33f6d8f, source_statement/8d863c878c6e61f4b958c89f280e7a0e6aec003fefb1f64c721a6b3f89c196dc, statement_hypothesis/9ea4f87b7e62c33045bf420becfc6f311afe1021efafd09c4af456ec09dea951, statement_hypothesis/a76a3539a422164b131685c48dd1cef075760d3f3350fe5792e3a95ab15f0e8f, statement_uses/3132caf0e9f649b04f4ebeb83db00c0ca9eed21cc2b2b9a00f986b0382c58306

en · unofficial · sha256:0b40a3eb701f7dea7adb848e1a8a09c4d8381d75e02c83feedbe2d7991e24114

nfim $p |- F/ x ( ph -> ps ) $= ( wnf wi nfimt mp2an ) ACFBCFABGCFDEABCHI $.

Referencing norms: proof_reference/13c6bd76283afbb6d5ec68b0ec8f9cc1eb5d66ca5da297cfcfc00c8b835f8394, proof_reference/1b39129387ff0455e2edb1ed618ef289b2c280b18aa7b730364ecbdbe744b51f, proof_reference/1e6d613385e7125cb1f0c24dcfd9c019182b9d956cadf3e2086584c00f43be4f, proof_reference/8c6646164c2177a575fc6144f7a57b6b2ab5b73855ecfec908b67839108a04af, source_statement/ffc66bce8fbcd37a2b8780c7790feffe9389e5e0c63e9c5b896d4e95ba7b1099, statement_hypothesis/5a3df42d0d7b460e166f159295cc337c0cdf41eeb114aee85d557318f933f783, statement_hypothesis/6e0e5ea95c18144f165bd8422fcd548054e8893388a4a6077284880e82f0d864, statement_uses/26006771bea7fe66e24f403d352dc6480e1c6f5abe1daa7fd7a9c0f2dbf1cad3, statement_uses/2cf2fed76e69b0b1ae7de13932d45203e9d0fc550ed10ce6d31c3c7d7a095fa5, statement_uses/338c558332f276ee948c5cb1419f9dd98a2e69f298030c18c60408c3d07564ba, statement_uses/8a67327264602852919fe5726e23304e4b0d5b8eb1fb7ed0deb43a1b3ebcc04b, statement_uses/9e7985e362f869d0066a13e93136dc0544db8d1046fed81e001ea322a88c5f2a, statement_uses/a885d1f8496a7c16a4d68d1fc107a7a4b3ff8357a84c60e710e2aef0c7bf3c38, statement_uses/daa502d12ab68c4cef9ab445548c0ebdadb9587d78f5a0eaab186929d11c1006

en · unofficial · sha256:31b03d0820ff6b90f52da17b81e656f8576fc5a5817577ecd1a05a4a58032344

$( Inference form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 . (Contributed by Mario Carneiro, 24-Sep-2016.) $)

Referencing norms: statement_comment/064739d902176fd87edba09a4935f964006ef159a7f6dfea812a26e65cb51167

en · unofficial · sha256:ecfbade1f0e9d87d9cd00f7b6192da0de5bc95e34a7d5a3f21248204f7a13ea3

spimw $p |- ( A. x ph -> ps ) $= ( weq wn wal wi ax6v spimfw ax-mp ) CDGHCIHACIBJCDKABCDEFLM $.

Referencing norms: proof_reference/04cdcc55c59565d15b08f1e3daed0658a5f3b088a0bd9f384c8d0433ccc9b2fe, proof_reference/47b598a57510d2aec893a46c6b0c28bdf9db89e7c4568ab9f06461b5f0eef137, proof_reference/57ee0a80aca7a667b475b3a36ce161bd56ab2dacdcf695c19730f7cd8f22d046, proof_reference/6e6eeaee2889bf740217d6d773f4973d239d2c4a7300a56f8236827ec3aed88d, proof_reference/7f8cb1bba592d56767b362cbde7c028b83588dcf8a20f10d0edcaccc9eaa228f, proof_reference/a20bade24b361c6bcb97c5a1d2fe9f9cb9bceaaa071b3aadfb408d0a7007600d, proof_reference/ccb6d3caac439bcb81856937e1c6e220593a14095e2259b8a0e6e18c750d5957, source_statement/f887997345e9aef278390bafc4794496b2f755550543b9651b3707c9e70cf5a3, statement_distinct/9c677188dd98fb731f26c93ae9aeb499100bf34e4b6b6b0f594dc2b722bc9d3f, statement_hypothesis/1a8d6fc34cb4ed9670745f0e049897eccd8d4a0834dee839617f375d74a5f133, statement_hypothesis/e56f6c49451bf2f53c8b2280c58955eda712f50b2989c63a6a0e0f63416b3127, statement_uses/04f35b9b87019cc416be421cac4df3ccefb08661a20d1e936b6a37033401ea07, statement_uses/0f70b9b2ea6e8e269a1da987e03b3703294aa181ed78976b93ab398fcbac6542, statement_uses/269bb616533491a2493cf10f7c9d669fd1ac25d246818724acdc23081446ef0a, statement_uses/45627cc3ade7efffb2eb55e5cb20280cf72cc36f0d54eb3fd253f6b5d9e99673, statement_uses/afde10e386ef480a793958ea9f4190a2e23a22b778faadecca351a10664d9927, statement_uses/b624317c63057599c8018a7695d95ab232ea93272fc09811c10e6ff1f7cea8cd, statement_uses/c64c038cdab4fc8ea6fa515c8f2b14a1872c1f186de905b763d8132be4349a5d

en · unofficial · sha256:df9b4a42057b039f1153813c4160227202f517ae22914b1df026b7482366bbf3

$( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# ZF Set Theory - start with the Axiom of Extensionality #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $)

Referencing norms: section_parent/6fe679b5fa1eefb4a10f272f82f9fc5db69951853604409b322fa06191784449, section_title/98da076e4506349734ec029a08ccef75c3709972f59798aac994180820524f89

en · unofficial · sha256:8232f55ef2d0637a1599338e9300daa97b058155c0d53fcaf03cfe21350575dc

mt3 $p |- ph $= ( wn mto notnotri ) AAEBCDFG $.

Referencing norms: proof_reference/0901e0ec47237ec9fb38290685733365144ca5421019e8b7c48d907f557d95ad, proof_reference/9e56a6b73fa6488aad5351ec393e845b67299d6a730a683eea0345abe6bdeb22, proof_reference/ca239d60dc9ace14e446d0d3495e37a590b8a7496a4507271ee3b409f3a003c8, source_statement/0c5b81bef8bb6877427ab9060706f2ed4117fd0ac0de6bcc6e309d392435298f, statement_hypothesis/bdf1b37ac2eeca47f6e5cba4a4a5def19bb612d435e4560f2054725bb2b8a79a, statement_hypothesis/d7e8a310c136715ea3d1eda2fcd0feebafa3431215216eeb6ca8950dc0312ea0, statement_uses/4f9d1839d2c47e593bee26a7d62fa738e6eba1c54574cbdb14d04cf2f7992ad6

en · unofficial · sha256:ec9b3e94f0e65b3a04a7de98c81f73cf15f7696ad8cf9798eb1f133c3a1bdd8f

sylbb2.2 $e |- ( ch <-> ps ) $.

Referencing norms: source_statement/a6c100d8ed204a46ab3a05b2717a828bde7002930e3ddda2b1e93f3273accee0

en · unofficial · sha256:b1ab5ae705edc5e73c8306c030bf55e56a9330431ab4822a6ab43f3cfc60f90d

imim12 $p |- ( ( ph -> ps ) -> ( ( ch -> th ) -> ( ( ps -> ch ) -> ( ph -> th ) ) ) ) $= ( wi imim2 imim1 syl9r ) CDEBCEBDEABEADECDBFABDGH $.

Referencing norms: proof_reference/5581a992a6f9d44e6a6718757198c787f29983d0f18d7ea199fcdb40d6f58bb3, proof_reference/ae28f253b2e5a5a296ef220dd648b33208354486ba24037b2cc246ccba5f2642, proof_reference/b2225018c33f3f725078b30ccb949597cd56b8296b5d514ba34b74b37e7f7826, proof_reference/ed925448ee3361ce092f1b1febecb4729dbe59fcac91efa5e196412bac529e1b, source_statement/ddedba16d007685dc43ebcf669d2f2bc45ea7da889a2af0883ac19726aaad5cd, statement_uses/4ac0f6d270d049263a103a2fd9ae0a25cc1d8363cdba9a0bb9b1c9055f38eb5f, statement_uses/4acff074207dab5958645693133365babd98dcf6f8fb762d6aba492e9a86df18, statement_uses/660415fabe58a768bc035f5db58cfc86eab40040e09cc3c51a68100fa46ddfed, statement_uses/b7c4f25ef4775fe2268a385e79c6ce36d24beceb82add12d30a87880803665d1, statement_uses/b9413a8b34636e84aead6c174978a31d0f9de2fb0435b0b7d758dad517384b9c, statement_uses/d1939330066753bdb88d495273be6872728d18b97aee9f8dd68ddb80a7247794, statement_uses/e8ed78d9929aeb335890aad98cd39eeda3daf2beb0323e9caecfe1b1b97ae505

en · unofficial · sha256:2fbd47eaf5f0c0297b4884e7d37d13e958d24cd8212ca3dfad7423c569ffe1a0

sylan2d.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/8890cc99e98840234c4a0b43ed4faae521f4f05134f7044f98f890eee815428f

en · unofficial · sha256:0be79945bd28b509ab21c8263c27f809dc73615bf6de36ce8aa85f65bec0843d

2a1 $p |- ( ph -> ( ps -> ( ch -> ph ) ) ) $= ( id 2a1d ) AABCADE $.

Referencing norms: proof_reference/70e47ed2f3b4338350d0b9b86a4c0b633f848f3b4fcb8383c20f93e81ca90c2f, proof_reference/ccb1f537554e0962a6453ff1cf97c3f5fa461ad01d1d69d0509694475ff0797c, source_statement/b8a09c7fd77f72fd337dbc0a845f05c15d95bce9e8c6b3df9ac0c2ec5c9c6fda, statement_uses/6b56b3ee3c7c690aff538fac26321f141f0e65a6abff7eae06c12d68f488e114, statement_uses/6e962738577621d49064f96726a5af7be115c9645bf220671301f8f262e835de, statement_uses/75977f127a1a7c5ecd37ae0802acd7af832225f2b2a0d9eef6a4e7647077f6b7, statement_uses/a3c984b79420e2d68167a08e99df76bd14f5807724cd37c42146f6df11c6e9f0, statement_uses/bee77ca2e38449d10ae07fdcd6492822d0b499d197f194e8909b965b1c1cf26b, statement_uses/cadfda9dbc6ea5384afdccbe59a981ab2c5aef5000d29873b4ae22b22817a393

en · unofficial · sha256:dfc95855c110386c4cac4f449608ca658ff9dee750daf27792177f72de872e79

pm2.18d.1 $e |- ( ph -> ( -. ps -> ps ) ) $.

Referencing norms: source_statement/976593fc277dfb48342ede19e703c8930e5176fd8de8a10f2ba7690d425c8ad3

en · unofficial · sha256:1533c25e94a7eff4ad59a06a5fefae005732474f0dd2936d73aa83694f482015

$( Deduction joining the consequents of two premises. A deduction associated with ~ pm3.2im . (Contributed by NM, 28-Dec-1992.) $)

Referencing norms: statement_comment/b3a12751a19f42b8b2cd086948f9917771bb057275c119ad3b6f7cb3a990364f

en · unofficial · sha256:52e7b8ae23453e630505444d0abc5904a6bf1efd3476058d1882187454de8696

$( Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/4ce239f7b32d9c378418cfa92ff88aac186431574c61b525cac13434c0901b88

en · unofficial · sha256:245117bef589f96c31866d328923fc93edb39624222e7754c30ec099c6368aee

com14 $p |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) ) $= ( wi com4l com3r ) BCDAEGABCDEFHI $.

Referencing norms: proof_reference/9385f5203f6fc0dd241937c7f9ea17e977f86b064cfef513557c0f61522e1799, proof_reference/cd70bb0b7c3d5d6438e5f290d6e6c0c3b101badadf840f8dea172695ca5265e8, proof_reference/d84df924a0b02eb4021f587f3adf855cbc7c89f2c618c63845e09a8b759b526b, source_statement/58717890c400fb995550a4229750f9eb115d794da71f19b087c42bbcc2d0a791, statement_hypothesis/ec2309cc79d2ccb450591155d5fb204714e001ca53a86044af70f0c552ef0d13, statement_uses/27b6c54fb00e79cc1b18bdf4fd00936ddc990a172a016ca7b740082dff51ce1b, statement_uses/2ea0297d3d4221dc3a06086f23844622674fdf8eeb8e7b953f4c8bb71dc7c251, statement_uses/53038791b954c529d9e44e862a26af5ce0e57e82c497a8db45e772c5275e75e2, statement_uses/6883e9b198ad8377f54217a3d0e412ae108c82aaeea6984bd75b9f4a9b6ad6a8, statement_uses/7b5ac5399de31dcce74e5acbef8755809e28a40927a13573ef49289b8d5e8a81, statement_uses/a4f603195698208a75ffb285a3f974225666000e8d53332ac5ece296f51cdb65, statement_uses/d3ed953eece23c96ea2dd185f415e2fa9e1ebf48d1cfa2bce7d8dab6567fa401, statement_uses/d6e5992c46b47d34e96d695d5d79361640f7336087caa10be1e6942eb21eb70d

en · unofficial · sha256:eaa36887c5d895d03ab1f09a96f20af74584dbd110495c591c063029490730e7

ax-6 $a |- -. A. x -. x = y $.

Referencing norms: source_statement/c4c10dc33d7a7ed0186ebebabac3e68c67a87c48a52a604fc5cc4546bc22f102, statement_axiom/4a7df611272df6be9947cd5a2c9e1bd4d288677abd60de127285453bf67f8b87, statement_uses/0258aa97e54cf495c7e60850230d15096500a97ac5e869160fcb8e0deddc32af, statement_uses/11b4bfafee2ca0e037606e1562d3527698337b0fc461d311e04a24fec38422a2, statement_uses/6ea6d4e60654dde15ad5eeb6144cc0e546388e8dcd9ac97ac0ff652b8992de2e, statement_uses/8ce82bd1a29752fbc0840a7b82df067c74709af4b82856345bf472cee1fac024, statement_uses/fea3730e1655a5c9356262bf3201f5fabd28c0fd8c633ff381ddcc84325d1bfd

en · unofficial · sha256:2a6489914c7f367cd2523fda7605d71658776e66001a9414ac0651dea26ab50e

$( Triple deduction introducing an antecedent to a wff. Deduction associated with ~ a1dd . Double deduction associated with ~ a1d . Triple deduction associated with ~ ax-1 and ~ a1i . (Contributed by Jeff Hankins, 4-Aug-2009.) $)

Referencing norms: statement_comment/fba9c8edb82568634c9df8a6f3837887343bc63c24daa950a349c20623284ef8

en · unofficial · sha256:d31b7c508b8e60dd5c4e5e580090cf7a6cf92fcb4695b3f308640fb8e51467e2

com12 $p |- ( ps -> ( ph -> ch ) ) $= ( id syl5com ) BBACBEDF $.

Referencing norms: proof_reference/0198428daf95571f72270680ee768f9684ac8b87bc75b4415fa13ef32f74392f, proof_reference/9b681f207b8ebba627b012a1fd2dd255bede86bab5c3fb95cd7e76db7f7a4aad, source_statement/d2647d99d3120e80d67cb49ba0ff3da0c8fdca64841bdd254e9365e716740f17, statement_hypothesis/c5924fc4c12f3b2c8d3394820cf391a1d19525df1a80d4a7ef29e9d88cd7eb1f, statement_uses/1b3aa422b32098ba6b4619c84e0a9f34507ba05ddbe9111c86e9714edf38f7e2, statement_uses/4fdb80c126ea9c4dc235e8552d22f494f4b9957820c5392e14b65488128d1e44, statement_uses/6dad1d6e0c6fa48e023aa1b19917734b84ebe263097c9620e827135ab207a8f4, statement_uses/840bad7e22ee55bb67d7b96c25af663962f580543933d10c284b9b2b58d5549a, statement_uses/98a5ad0edba907e0cbc624187d5622f20e86a8b69ef9ba6d6e1add83cad72c25, statement_uses/e10eb8bbf8030a3f775b1030a616459387f35c8efcf631042280c08648425c64

en · unofficial · sha256:52e4da1dad7da5ac6cf61a7a8fff018ae37280b21b8b077f28d5f1d665436ac0

$( A closed form of syllogism (see ~ syl ). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is ~ imim2i . Its associated deduction is ~ imim2d . An alternate proof from more basic results is given by ~ ax-1 followed by ~ a2d . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.) $)

Referencing norms: statement_comment/51cbd10a8cad3a3c9330277ed67061ae27813a67fbf7838b0d0be913978f2afc

en · unofficial · sha256:2983a98b281d5eec838f68a31bd6fc91ed31d2ccb97e2be2faa795613d5b0b4c

3eqtr4i $p |- C = D $= ( eqtr4i ) CADFDBAGEHH $.

Referencing norms: proof_reference/d10b0a277fbc0391c33b202ce52b384111c828ec1b2cef43d63eac6c4e1d7e12, source_statement/9f48ef413eb5226de5fb4f46dbefe11306db3b58e619d28235d4ddfdce52e123, statement_hypothesis/9693e86b596db8633ae7a48e754004b84b29156fd98ae339541a3718523a359e, statement_hypothesis/e12ca4c5416c36882cfecb74509cb20b75fd7a2d0f0ea1ca51a2432316cbf4b0, statement_hypothesis/e7d48886fc0a5a718fde759ac7bc6141a4d45fc73fdfd526faf745b0fe9c8111, statement_uses/461f6d69c2e5d4ba375a6c5c80211a11636c7016000660eab760cba5ec6f1f7a, statement_uses/468867d44ff625475e439fa5f7051f859fcdab1ce641a5efa34d392cd6a615f9, statement_uses/8391eeed5ed407c6176b5eda2cd53602cd929612fdbb6c25ed9bb33109e2d0fe, symbol_name/f2d26ecc4d4965e4023cb8f79989aadf4bf7580df2e2e90a6b8664deb4f9a24e

en · unofficial · sha256:84731b060a4050a418194b43bdea057d637c1a0d34ef8b73731d6a1eec17832e

bitrdi.2 $e |- ( ch <-> th ) $.

Referencing norms: source_statement/80cde81b4afa2aee151562850ab47e30ca0fcd6811ff2d8dee60c22a63c95357

en · unofficial · sha256:7f9ce8cbc10a2b863c854957a72c8dd7c3d327ce824710c0635b3e45c4e3678e

alrimi.2 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/fba55cd2ee81c2563a08097d214df0f62fc125cc553819f05ed55b5c39097fe2

en · unofficial · sha256:10297967b221c7f8e23b393960c6eae507e28d2375dbb99de7647f5c75e73067

nbn3 $p |- ( -. ps <-> ( ps <-> -. ph ) ) $= ( wn notnoti nbn ) ADBACEF $.

Referencing norms: proof_reference/31d1292d008d772b5c6b666089c3e843b5d7561f2fa222384396f3b2b7e13d1f, proof_reference/befbfc49b9c65c483cd6d7aa366c94ab10851a817cf31f8b18b6318a930e3f5a, proof_reference/f64525f7ac873a360322ca6987694f726bd1901fea616aee7dde03fe1c8e3ca1, source_statement/5574fe4b1763a399b8952ef72d99368b25742fbe492872b3ac0e424644c2a414, statement_hypothesis/a408736fbe85ec770b57d49763724353d5b58d6cb85218ec1cd32a1802ce95f2, statement_uses/2c6c41913c725d5d39439d74c67b116ea44e3fd7ae43bee6199ec506b131c6c3, statement_uses/3aef6a5f3be6dd7321d46f54123b0d6d02e37c70f9b6818b85a13a009fa85df8, statement_uses/75911686fa0064b82f7ef2b0cdfa3f71a916dd548e48e1e3b869ea918424b22d, statement_uses/d13e94b36916b5c28c980300428e488c3bef137857b5dea65ad496490b957740, statement_uses/f3f688f4fae83d5dcb8ce729fffa30e97ec2ebf8444629cfa2636738a237eac2, statement_uses/f45170b50a7a2bbe9b7a269df7f4e2663b30e35bfcf64b57ff66b8de58813fca

en · unofficial · sha256:5a884ce2006f7e283d2cf495f46d298e025096e9755d5bdc34369bb1ddb3638c

notnotb $p |- ( ph <-> -. -. ph ) $= ( wn notnot notnotr impbii ) AABBACADE $.

Referencing norms: proof_reference/8c1a424dbe9bcd0bae2eb093e531635b071b642cf4803acf1c1e71ec3672913b, proof_reference/9799fa58fadecb1107055a3fd0258e6e7e534d20065f41d2ada8b3489a00ba99, proof_reference/cf38b220bf93b41cf39e4311b02d7df0f9aeb743b44d62a5558d476affcb430c, proof_reference/e93d87b635f0f8f16ec0578138e0771445d228291eabe8c3176e64ee225f25a4, source_statement/62fe3cc7e6e21879b67f418ebb2adbf69c2256093023602da6c20fbb4ff7889a, statement_uses/527bcf2526c3589ec880977cb73e152f690c73f0811dc0d875d0c2154fe3ff24, statement_uses/5caa8b101ba824450ec4519f13b5566c7b8f423087b1a302ab1118d464deff73, statement_uses/a00d45a4aff2e173d60a78fd113eee1fe49446f2f0bb4ad89a68286509dfc176, statement_uses/bd9bcae3f5e4b60c619c95a735b4cfd90dccdde6f295cc6f51e6a70415c35a7d, statement_uses/ce056c5702d3c11703033648e41353f29023a1f2d04146c8d184f9f882d884f7

en · unofficial · sha256:6c0acabdb6eed43f7ab1da494197b02b2683051941b109f68fa474909811e243

orim12i.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/2ab2195b34f24021dae43eb387ef38d931df06a3e1121c359d2e2c77e1e023e0

en · unofficial · sha256:58a1a1cc4199d7e3f97361769cccc4867194312109f3d10ab31040086cfa49cf

$( A negated syllogism inference. (Contributed by NM, 3-May-1994.) $)

Referencing norms: statement_comment/b31550e7c59391cfe7e6f3ac53f70aa9b891705fecc8b6f555c01d43c8dd274f

en · unofficial · sha256:1fb82f002a9cbde76d2fb6678718f2b7666918a0f23dccb1951a090e312ae19d

$( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# ZF Set Theory - add the Axiom of Union #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $)

Referencing norms: section_parent/20bd31e105f6aa7066e63ed1ebf2f259755acda6155702fe50dfc1d8e15f7d37, section_title/7b76769fe5c42261fa68a3659557416389542a00f2819c3ddc655d76ddfa53a7

en · unofficial · sha256:68b5ea340a1efa569b692f20bb5fc298a52ef706235f74088ffc29b75e2ecbb1

jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/e71e57017150a6b182c6625b885437fbaaefdf3620a9dd1d95d6ce1a28a756b6

en · unofficial · sha256:224afb5db0e6d394e4d8aa198a3f7e5e20efeffb1d25e97e310e512edab334d4

hbn1 $p |- ( -. A. x ph -> A. x -. A. x ph ) $= ( ax-10 ) ABC $.

Referencing norms: proof_reference/e73efa6688fc97a90f306fb6dcfb58ae2aaacd02df6fd153052a0e96146435a1, source_statement/25a7db976216255c703b213d8512604e3d4ad86195206168e28d2d67c46fb3db, statement_uses/0fb455cf72214cb3983de10e32c53319886c645e4d057763c004f2594c32f0ad, statement_uses/2384080d36f0e16cabaad8d00ba9dc0c6ecf95375d3ef3ff0b9c815b8a215b27, statement_uses/2e3f5b4ce3ac3a18527fd870695ef3e9615754d6a6beb587962a4c5ef3c10a6f, statement_uses/44724caaed6390a08f95d8bc10cb4e7434dd252be4b23d7ea8129b5c58d688fe, statement_uses/55f6cee6c4d80099c3d08fc095b8f438ebbc4edd9e19f99d48e91cda90228372, statement_uses/7341bd50d4bf9ff13e39f40168aef064d00605764e65dd09f853d9c298dec8b2, statement_uses/8754568e4046c20127015f5d8f50031aaebfd6a2bbddc09cb3511e847b198699

en · unofficial · sha256:6a55538e9accf31074171245f67b3d71f0a12547d29686957bba098468896eb8

$( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as _Frege_ in the literature; see Proposition 2 of [Frege1879] p. 26. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by ~ pm5.41 . (Contributed by NM, 30-Sep-1992.) $)

Referencing norms: statement_comment/b95cb0103330cc1144d68291e9df5f42c35f7c9cf76e4c5af210c440fb3ddab5

en · unofficial · sha256:62fc11806a6b11426904c195a4e9613e7c4fc501f0076bd16fa1076e824c6f1c

$( Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) $)

Referencing norms: statement_comment/3f43f04c2d03c706c3ba49b6d3d59437bb4186a2563f1087ce304d4e88a1d881

en · unofficial · sha256:27d407e4d7b50c256b14fddd1d0ad73f25ad2b7c52dfa2e621709efed061289b

birot $p |- ( ( ph <-> ( ps <-> ch ) ) <-> ( ps <-> ( ch <-> ph ) ) ) $= ( wb bicom biass bitri ) ABCDZDHADBCADDAHEBCAFG $.

Referencing norms: proof_reference/0215844782b99f57187e06d0a9158d4e16f71be2edcda18edf686f53c587dbf1, proof_reference/98005bb292a310412b171c280671c010a70924e21d58ecb46dc788ba230365fd, proof_reference/a14f44e2777148ec42e45b142c093351bc8ba1b728a2cf3bb22d01d07e59df06, proof_reference/ab19cb26c4a800bdada7615ecb27723ce1edeb757fdf0b09112624194835d19f, source_statement/d5fa28eb3c9b12a2a5e53bc530c2c89ee802244427fbaaca3cfa1542cf44481d, statement_uses/18f7eb55cc74fd3c89b1f4caea461420a2c113aa56e37daf296cb453a8560e7f, statement_uses/1f71d382dd9d696da6eccdee83e8b570391100536556667bc4fdb9b1be11fa1b, statement_uses/3b7f45b65a559caddd0784f7a8c053bf0b267cd68e747d3f486b6fb852d042b5, statement_uses/70c133f098f06cb6df2d98c22d36331d0d673d22bda4d4c82709d65921878426, statement_uses/eceeeb3bbaf83eda02d2303d8b6b568b9d09f0258c53d82fb506b64ec00f78b7, statement_uses/f090c670bd53d6569c90b1b1c786034c3778c625ad0b1645c10d2f55638a95d6

en · unofficial · sha256:96b809390e23c8b3404baec08d0b734f85f2fcf4d32164a5d4490b1caca745ff

$( A condensed form of ~ ax-rep . (Contributed by SN, 18-Sep-2023.) (Proof shortened by Matthew House, 18-Sep-2025.) $)

Referencing norms: statement_comment/a43c45b3bcaf8485e9d80705342852b3f50c1ec910809f10bb02df3fbee98065

en · unofficial · sha256:65f8c7e98c5e9794d07aee7d217820fa43c8faf28357d2f458440c57e9b57fa7

$( Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) $)

Referencing norms: statement_comment/0e83a81d48bf209e160c1a13be378b4d577117abe07f8f089c002fedbccc9fec

en · unofficial · sha256:95512dc4c0aaf18041740b2f635fe3f95dadcd973d0598eb0219c06024eaff2b

$( An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.) $)

Referencing norms: statement_comment/205ca4554cf794c75c94905e0defef8d16036ed7b91fe37e79a2a02f24a75e3a

en · unofficial · sha256:7f6afcfeab5f0de1eb8edbd48e292d971dc15c6b13b59076556491b41acfb994

$( Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of ~ eqcom . (Revised by Wolf Lammen, 19-Nov-2019.) $)

Referencing norms: statement_comment/81e27678ad96ada89237c221c4f5b768e196df120bf899961ceba70d32210ced

en · unofficial · sha256:a348ef5acd468d63d6d36b6e297d35e233349400e6207afd4a76b330b0484fef

imp31 $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $= ( wa wi imp ) ABFCDABCDGEHH $.

Referencing norms: proof_reference/1f25967fb23ceeb4a771636b7147f55420ff6673749513fd4a9b1d0cd5cc1ec9, proof_reference/872a79e905f0e419602a05530b3f25c6f7e24b94c2c3199a2aa730938ffe23b1, proof_reference/d806a7ab449b0d85e8d33d5de4731edb0bdeb483401ab402d4f711e553a14e9d, source_statement/68e234b79102c95d9f7a9ef62a70c796a998c19c53a4e304d93aefdfe7e89e8d, statement_hypothesis/292b0fdcab3d22c0fda5c476f668d55f83d300b4d2b6b1aaa308135633005931, statement_uses/1c743a367d8bc358b1184ededd7468b6728f578a998aa6a76d215698651c5031, statement_uses/2c8ae786957784d698ed49a3d20c69aaad4db8693987111973cb8752c8985f27, statement_uses/6450cdf15be9b02b118d0b0148a2dabecf65952b51a75af09d69e224fc8601b1, statement_uses/6bb1553076be343dfb28b572c20504b280b144cb3b473f8f9a587a274307af80, statement_uses/94829cc656fa38308b743ab79aa870caa4b52f859006b22f4e5a7af8af6bba63, statement_uses/9b62b4f3e10aeea1f884fa8bdded615a254148632c074cbc00f0e49ac49c432f, statement_uses/a95245246d343ee2866d7c6124a0a6a7fe67e27d313bd0f374aadabee854dd58, statement_uses/c18ee82d60863208c74a66be19e351b96887a73e73ec4534dc561de16c988d74

en · unofficial · sha256:5090f1440de6f66b49d4017989cdf1abf5444243820f89858d3d4e059afa9436

biimpd.1 $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/ce5c34f5cc8516a8d4a2c9133036e90d53c8e15522418618b48e21e9454f23f9

en · unofficial · sha256:220b570980c99263b769c7567e66a5d975fa91becc134502b59d0a48fa38c8b5

cbv3v.1 $e |- ( x = y -> ( ph -> ps ) ) $.

Referencing norms: source_statement/bcb394eae46b29b407fcb1758c0ece04418655f3a27db82580ca8503dbb46bf5

en · unofficial · sha256:04cb078abc69ced20d3e86430fb292a5b80a57e874dd0b2640ee498e1a463fda

idd $p |- ( ph -> ( ps -> ps ) ) $= ( wi id a1i ) BBCABDE $.

Referencing norms: proof_reference/41db9f050fa0d84b44e4eb3a395152a69475952bb81c74e476da0d4bcd6bfff2, proof_reference/4fafd86c2eafc5d093f390d86b60a683b55868ebba66acc03bec5a0a31684c17, proof_reference/e8db7be1bd3528c280eed6769a36acbc7f4ce176079df2644e55981e33f9c28f, source_statement/002f985b878de4220973d32b28bd1dd611138b8b92ae3698ffb691e78ac5a2ac, statement_uses/163478f6163f2e30a00052f900ee6e2bfd6b0c1d65d622eb7c073569919148ad, statement_uses/17f5b82e38ab9488ac00ac344376dbd1b8b4e4fe07098a22c1bdea4e0d8cf081, statement_uses/3b400a290a82a8a2e9729f40438077e288ca5ff7298df57d13689f782ca51e71, statement_uses/501119cbef559d410a02234db1b53593b21d6f1292fedeadb9f278e97f44371e, statement_uses/5416779975a5662b9b9b162db792df301b3d92c9374196d2c2e154091dc22596

en · unofficial · sha256:0a105095f59b1884b21f5c5b5291bd9e0167841ed9f66402da39e390ea1ca64e

$( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)

Referencing norms: statement_comment/d794ac076b170ddff84fe3243cf0b1b21a93506e7156d803208765aec5b0063c

en · unofficial · sha256:eaebe8623679a5c4a94c0145e858441d329168a7b47745d7dc5595126f6ee443

syl8.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.

Referencing norms: source_statement/48e47eb8397c62dacf93c984bc642c34e0ea7dc372876865b09571a44745120e

en · unofficial · sha256:e964f50f45c18b8b9a0828d41557d289775da717005726960778c0b60e163397

biimt $p |- ( ph -> ( ps <-> ( ph -> ps ) ) ) $= ( wi ax-1 pm2.27 impbid2 ) ABABCBADABEF $.

Referencing norms: proof_reference/5141239e6a729aec40f482242c321a742e4de08ac90850eb422c46b23c7c1c68, proof_reference/c424915f452f5116916965b8d99d1f3cddb93bd8b596c66352a704fa65d6faf3, proof_reference/c94d7baa48588ae65149aeb4132a27e277ef00afa1cd21b4413634e34f0cc6e6, proof_reference/cff0c70f71d48cf31eb11211c5cfa1f25d63f41582b35e2e68697003ae09a816, source_statement/d2e227c834a4ebf2df73e9c62c7fb4e1d9cf2cb0162f98b5f19ea9f9fd7572fe, statement_uses/18d53b4236299f7902b5d01f41b4634f755cf2f0cfb7ff5f7ece25e3e0808e42, statement_uses/1deb55f2cab766cee1e1f6475bb89480c801475ca98cadcbf7133ccbf82c25e1, statement_uses/3dee1a1625ffa8a08be675a516904e191d5334a24842f35a833e06bac77e0291, statement_uses/91b7d24cca82b34ab2ede04bd460224a49fa137d393a2f79519985cef7c16703, statement_uses/c303c76e94f797c70745d01862a0a09a78504212d791090747a10fc5d982f943, statement_uses/d280bac703b808bb7a14ebec7d12a02abdc0c277d339954a207d1ba6207bc249

en · unofficial · sha256:9fe0d5d74174782e9fcc23d287f072af498327eab3fd6a3c134e9ae94c2dbe9f

nfex.1 $e |- F/ x ph $.

Referencing norms: source_statement/a9f1eaff5f07f325dd91e3503c57fd0902e7f6cf4f9ea99af6b08ff17df89da8

en · unofficial · sha256:4ca5e6f2be00bb6d24148ca46c124f2358c9e9d5def9f8a34d47887249b19b3e

$( Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) ~ df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.) $)

Referencing norms: statement_comment/80b959ceab3af8861104bd762fac32778b6cbb3d31c56a6da457049d2ed1e696

en · unofficial · sha256:8fda8e3cfce0e2d4fd6ff1b1e5e85148ac9c4b7fa9f28d76c31ba15423222998

wrex $a wff E. x e. A ph $.

Referencing norms: source_statement/9dd91a5b4973364a59cfa2f3dbeaf58d3107401cd5bf359da7558c4f6ea7b2e7, statement_uses/914404f08da475b27baf9ceedcc886891955e43b1f0da65f34590d2b577be277, statement_uses/a593eecca5e1c20697f4a04a7699c40eb1bc30fde3e1f80e44cbe13fe7e86d22, statement_uses/bf1076a15dd53782c17781a5bd24a48d96db334a47f468b5d7ddae7d1fced218, statement_uses/e17ae741e3693919b8a502e8f413cb44b31c37e6346c6fb7301b192b79e9c66d, statement_uses/fd9ca3ceb24e94c431bbaad5af93f57e546d62ae19cc59aec8c00bfeb492a89b

en · unofficial · sha256:d437c3aada7c9365e1bf5c5896a65addd005cba838f1e4144ea1927ae6b19e6f

jc.2 $e |- ( ph -> ch ) $.

Referencing norms: source_statement/52e947b02e3d20c20f4af83ff46e50860f65c2e996aa29a173d569a2c4c8a77f

en · unofficial · sha256:3142b2cadb331cdf0c08d739c70a6caa53210b5ce02c2704a8482902349425a1

$( Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) $)

Referencing norms: statement_comment/8b3885cdbea88700415bbe108e379661b616aff8054ddd28d92cf11f8558e46f

en · unofficial · sha256:8cb2ed80cf011409788616ab8aa0ac25a14cd3c054474053bf88631c4157d55c

$( If ` x ` is not present in ` ph ` , then ` x ` is not free in ` ph ` . (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.) $)

Referencing norms: statement_comment/ee51491931a2bbc81ed07dcd949af5cff76c2e20cfc2bebadec55e167ab56355

en · unofficial · sha256:63449b62d4655f59c9cb0f00d2d809262ca6dadff98d60b45b3eb11a838a7238

$( No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting "hb...", allow to construct proofs of formulas of the form ` |- ( ph -> A. x ph ) ` from smaller formulas of this form. These are useful for constructing hypotheses that state " ` x ` is (effectively) not free in ` ph ` ". (Contributed by NM, 11-May-1993.) This hb* idiom is generally being replaced by the nf* idiom (see ~ nfth ), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.) $)

Referencing norms: statement_comment/18f69bda69628948e11c7d6b6a5bdcddc1f768f21f3e7da6c99bdb7d3ed06efa

en · unofficial · sha256:1630f6201cb88c52131a35774b49d539aeb9fc596e3631a62cbd8fdcf3c6ea23

nf5-1 $p |- ( A. x ( ph -> A. x ph ) -> F/ x ph ) $= ( wal wi wex exim hbe1a syl6 nfd ) AABCZDBCZABKABEJBEJAJBFABGHI $.

Referencing norms: proof_reference/069d811d039846df1b46b70449e7375d4bdd91fafc178802d18e2e1a025c969a, proof_reference/41e0245645d3d2ddf92b25c09ccac3dd705aa29f55d1b175b8f3f59ce4d4af9b, proof_reference/7b7cd9027a7cecad2bcea562e3911867a36375f0139be29cbb902cbdb0c8bec7, proof_reference/a0aeb20f072f38a68ba22cf6bfdc6104b0e6b26dc51d55689dabb48e14e47eae, proof_reference/a881896c701428c194ad7edc253ac9b0c08c6152bf9a2f75862df1ee243fcc18, proof_reference/e4db0bc50e6f7a8eece86cdb5197b859089d8e097d5414c62e473b8345aad5ae, proof_reference/e5024fba45d04df82a089f5288e11b24b8d816afe29c7b63c3824adfd880bc1e, source_statement/d7baac3c2ff2f3cdfee0c5ecca15cf3eb6c633cd69a5a3f90d76368729ae6780, statement_uses/07b8ffdc6f4f44fb846898fe44d4f7270083c2e0225d57918670439b9d0acebd, statement_uses/2c622283fbdb1cf5078c303f70e02554f02a4f8e6b073688cec941e48d3567f3, statement_uses/36f5ac1224a9e1176672bda4dfdfe7decf8812c2b67783ddfa333c4f4e4d7e8d, statement_uses/400fa5a5d0a6fcde15fcc1a31fa9f603d018dc33352cdd50e0bed0430eb4d319, statement_uses/51a82cfb954a43edde114a57b7dd73a0e15780eb752935e1d02426629f629335, statement_uses/848bc8cc2c0ea0d74775a68a4657467bb7cfcd1e97383b153f74e6be5ce1c761, statement_uses/8d09c0827feca56b4c7311dbe88f02a15929122aef19bc5feb877cebfce59280

en · unofficial · sha256:6921d3318476cfe57cb96bb5a9ac6bf093f0f81b7b32f813b1202efd75d6f5cf

pm2.21dd.2 $e |- ( ph -> -. ps ) $.

Referencing norms: source_statement/74ab6127d1ac289256e34e9f67da68cf4e5bdbdf80c32db7a7df2da2484a888d

en · unofficial · sha256:a7f26646abb5ddec57ec00326cd193c503b3f2bb91267631998a21211388e67f

mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $= ( wn wi biimt con34b bitr4di ) ACZBCZHIDBADHIEBAFG $.

Referencing norms: proof_reference/2a1a71066d9686ce8a14a8ba613559bd143ae36cd2985a9ecf6e618bc463c428, proof_reference/5d579252a1bbb009f0f94694273a1a471a119771e8850b085465df219ed6d3ed, proof_reference/80661ca9f113a360d5e92beba82c513f222fce89890cc6ea45405dbc55ee6b1c, proof_reference/bcf7515e81a4723f503166ab7407d85aee8ce4ce36d250e01ef34e148339e633, proof_reference/c187768cb1f278535bfdf056ee89eadd83406adc5b5e1dac7bd5d3d4ce770d3a, source_statement/f8228be7e72b541f1a1f842210ef760c40dcaae5cf8675ac60a301531229cfd9, statement_uses/011468bfdd4860213fd26f9d94c166e5cea7bda99b07259f62229df53befd5ac, statement_uses/1939c847e808656776ee000554510fdb79fd08e28bafda0240620f92832918ef, statement_uses/46ab8028e631ff281b291fe6f41cf7502aeabb6abbfa6caecaf938d2769c8f2b, statement_uses/5b0f8dfdbed48e02dbcfe313ff2bd23b7cfb1e60160a0d77ddf8a782125f467e, statement_uses/81cf9db784b9c9b93f2b331542d5a3b07e3d053494749dc9ddab125e35744999, statement_uses/96346011a5f11b89e17fd6657bd1a8701841eac2034f7accbe3cc65022d3cfcc, statement_uses/f0f3ca2bc6f848541ef4da4408d7475602feccd210749102110231e6cc02691d

en · unofficial · sha256:b7134c095238507fd119c5f22301e34855c959f81d57f672c8bd21eb2abd02b1

nfnd $p |- ( ph -> F/ x -. ps ) $= ( wnf wn nfnt syl ) ABCEBFCEDBCGH $.

Referencing norms: proof_reference/2bc558fd45312a1444c3270778f92d3fa67146a485ca69050936fc4c7bcabd55, proof_reference/4cac2c6dba3407a5e267b9b505ef6d515570232e4fd331e464dbe698688318a3, proof_reference/98edd3af96513cc88c92eea2913d10f33852d92a772efadee9f8549dc4484e92, proof_reference/fdfb70aed67c5ed0d75dcd99b1d23e4861817cc2ea5c34be4665a79e5a3d6eab, source_statement/91bf8be40e70538f123636b968b62c4ad3dbb2879ffad9e03e507b09d6ed79d5, statement_hypothesis/3b12a44c6ccc6f909c8c7c85b51235a3af74d30f2c94aa981d9c994c3bbdb357, statement_uses/2bbc035fd35e26711a35c6d84528fc2d205231cfeae0d85ce3551d7ab7579f7d, statement_uses/31ebb9bd72645b1507fe75ca3b19a8a6e36127e48d572b319d4fc3a64d40176f, statement_uses/37ff0dd3deec35a939d1213effbf7145fab04c34e611feceeff428d973de8b13, statement_uses/3ca5a4939021dff23caa5bda0937757d3a099738f10dafd5aa876f2dc1dcd367, statement_uses/53ede7d5a0c78f2e741ad31ba202a3505bfe72c7bfeda9ff8c4f3e9fcefaec76, statement_uses/9a4dca16bc5fe226097021942a6c5e5bddca57d91b5320f593163afad0c1c017, statement_uses/f6589c39bdaf60c0b4c7c48247178da486d782115231d99202910fcb2bfd5a4b, statement_uses/ff0c2385f87ebd92490fe66437d8e24871a78075a61b4fedce8002e39115a860

en · unofficial · sha256:0d7422c20db6c5c25bcaae5400fa0429eea72ebeb79ae6ddf60107fbbdba0845

$( Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as ` A e. _V ` , asserted that any collection of sets ` A ` is a set i.e. belongs to the universe ` _V ` of all sets. In particular, by substituting ` { x | x e/ x } ` (the "Russell class") for ` A ` , it asserted ` { x | x e/ x } e. _V ` , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove ` { x | x e/ x } e/ _V ` . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system, which Frege acknowledged in the second edition of his _Grundgesetze der Arithmetik_. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ~ ssex asserting that ` A ` is a set only when it is smaller than some other set ` B ` . However, Zermelo was then faced with a "chicken and egg" problem of how to show ` B ` is a set, leading him to introduce the set-building axioms of Null Set ~ 0ex , Pairing ~ prex , Union ~ uniex , Power Set ~ pwex , and Infinity ~ omex to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom ~ funimaex (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types". Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ~ ac4 and Cantor's theorem ~ canth are provably false. (See ~ ncanth for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ~ ax-sep replaces ~ ax-rep ) with ~ ax-sep restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic", _J. Symb. Logic_ 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by ~ elirrv (derived from the Axiom of Regularity), so for us the Russell class equals the universe ` _V ` (Theorem ~ ruv ). See ~ ruALT for an alternate proof of ~ ru derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ~ ax-13 . (Revised by BJ, 12-Oct-2019.) Remove use of ~ ax-10 , ~ ax-11 , and ~ ax-12 . (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.) $)

Referencing norms: statement_comment/c832c071733655550746c5df514d7738c2c8a25ae1bf67f01ea6fea7a586ffaf

en · unofficial · sha256:6bf4f18603b0fc25a33286633735b598bda1eb41711b516a184170ab2d1affb7

$( Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)

Referencing norms: statement_comment/a2e823c43040c67fbbfdb660e63c89de36032adc92cf61e4f21aab0aaf024000

en · unofficial · sha256:2cf42909edd9103440da2d36a5fb024e551a387063b2c2ca54465d5ca9f8e86e

albi $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) ) $= ( wb wal biimp al2imi biimpr impbid ) ABDZCEACEBCEJABCABFGJBACABHGI $.

Referencing norms: proof_reference/3ed57158f5229c8010007eaee284737a0168e6a8a8e4293aec15aab79d94500b, proof_reference/492d6166bce7a11f76cfdd1c8aef5f4183ee59caf2fce7ab9125f5d5b5481403, proof_reference/6a78786894960fdaca9f8b33252d161a16c18f4452413cc23ed76d244be12781, proof_reference/a6480271bcb9a6fcbfd3a24c96f7145d86b9250dab7fc9165238bdbccd5eebec, proof_reference/cb4831f7d047c518d8ed19c4ea2a041bde6eab73ba3919d80dd75edff39f1370, proof_reference/e2d010d7f8815667ee0808acefe030f042c812f20082fc5fe958c3b15ae273fc, source_statement/94ad961957d5435e33977d47e9e344d805aa99ceaf5c68dd4b804799deaf700f, statement_uses/3dfcd531cca92e039b981fc09933b2acd4da71f462f37b31e2da201f2aa5db94, statement_uses/4b0f007cf4459b136909e8cdafee3d1cc7a62f3c684a84dee8bf346cddbef554, statement_uses/4b4c7186be68566326845e6253839a64532081e8d6b36d4f71989326b4800809, statement_uses/55336b88c83bb1618f61d7e1b28711b5d8669ea391bf719755c84ffd7b199627, statement_uses/ba76f1794c1cd9a8c9ab1c9d88c8608709a62583c8bc4d1bb4e47541fdedf596, statement_uses/be60e5a6a9364dddf29e5588bc3bdd5667846e2e51aae98bc70179ee67c69a3b, statement_uses/e5f17f8f0abc87b8153db906747ff9414cfda3fce57dd3a04673fb0df9d45701, statement_uses/f51debd8a6f60963c36afeb304640c08c0369d01dc99bab5967335046c400971

en · unofficial · sha256:61b5ba7666a51b43a7a1c13f9e2a66113205cfc80daafdf55fa39d7083f3052c

stdpc4 $p |- ( A. x ph -> [ t / x ] ph ) $= ( vy wal weq wi wsb ala1 a1d alrimiv dfsb sylibr ) ABEZDCFZBDFZAGBEZGZDEA BCHNRDNQOAPBIJKABDCLM $.

Referencing norms: proof_reference/1d28dab97afeb10b865da9146a076fffb39d7d1343c709d8950377bc5a289234, proof_reference/24a28309c40c2c3af8bba99604551ae792cf088c0d9ee33bbfb74fd3587d6116, proof_reference/369d9486c40d9914d499f6c92f57c20538cbefb126787c785dee13234459e5d2, proof_reference/45ec9e9a792fe6d9213aab1a5c322c096768efc30489f6c472e4cd1700d0af47, proof_reference/46f623711bf6b264689b927efb64babb0ee44c71b4b170f7170f1af677c94b68, proof_reference/53d9eef52a3ee74c2916255ba4b9e8743d9e787068e88d9836d3a1af09854700, proof_reference/7761b4d8a6d0c22c31fd6c04a7a6d380f75fec1bff14a43789b0f3b05bc3f733, proof_reference/7c00a1363fc390133d1827ecdca0877050a23e85ddc61564cd76a6893bce2a06, proof_reference/81bf9c59b8c29ba1728b64c4ec07c892552a6c78a5878b1ef492e268e3f9a6e6, proof_reference/d109b76936c5666f4687c71e0a742c845d2b9e389c4783068ddcbc552ce8b03d, source_statement/dc614c5f40c59c7243576946d24fc646ffad32418a3904d6781d8d0cbe01a9fe, statement_uses/0a58ea940f086f88fd5d9cb70337d7a3f2d34cc0eefc377a997fea92df42b321, statement_uses/0c4ae86c237b3d3a02795b63892219a79b7388e19a04e8f627ee8dd43e67196c, statement_uses/11eaca98f3c52340e0c68fad99060b2f59e9be481a3fe052ba41155e8feb7f19, statement_uses/3397bb74594ad019de1b1742af58a713e7500a6ec2e00f7a89e39808cf39f591, statement_uses/425c67f3b36af30e674f6db9c7e61f15f2c428df87c0de9e9f802f2fa9632fcd, statement_uses/768eaa08b4b9a5174050447d7a9bd3b739142714d967fde9bd50177b6c535b40, statement_uses/8ab0bfd96062a34de318f91912685d57cbbe7bbe992a4343780e007955aeabbb, statement_uses/b01642f2755604902f5798bea47141c3eda0dc3bbe3399f5f402ed4ff9b554b6, statement_uses/cbf6d764cd90f690dc29f3e03c05acfdebe448c79db49482ed7066b3b52b9ce3, statement_uses/ccdcfdecce5d963de79bdb7040855878ee2eed5e297d66e4c001c173d39dd31f

en · unofficial · sha256:c4a4094710ad02a6551d6763d3716579433d8a6aadbe921040c18c4d7d93d066

gen2 $p |- A. x A. y ph $= ( wal ax-gen ) ACEBACDFF $.

Referencing norms: proof_reference/eb4357393ecfdd3cd759f8208b7a715ac74362bc50cb8350bbb30053c0de008d, proof_reference/ff63f27e40ab2d9806d617595213c2cb80476d2f448b63bc9b3c860b8e713e02, source_statement/e280882cb7a078b9fea588b35895891598f5434297d9a3c3fa26831cbd937723, statement_hypothesis/3b31a87785aec9077fc6a820a57aad59cb12cd316560431334ec5a9ec7d3ef2d, statement_uses/39bb98be3e32bb032616c2e602bf733ed6ff5f6e40c015392075a578ba7fbe4e, statement_uses/4a75fa2d85e60a07f015d24629a5b1f0508c79ccb3510f760c0abab36edee60d, statement_uses/5749431b79f259056a39e844da2bad4d46cea787790ebfdd709a11cf85e3e38f, statement_uses/beeedb543ad3c759ac9a77a7c257e780a24ecd8c449f26340a24e87303b13d13, symbol_name/92b9d395b272467ddc58670e1f437b800fdbfe84df999b2f3552a3b45e6074de

en · unofficial · sha256:59dcd4451d948dda41509404a4679dbecb7f6010a37da4e67d44cccc50d51726

sylibrd $p |- ( ph -> ( ps -> th ) ) $= ( biimprd syld ) ABCDEADCFGH $.

Referencing norms: proof_reference/3ad7b7b762d13093f688e792cd7ec00a208f2d2cbd363a323f94e8e9a1a14750, proof_reference/e969e590c3dd08c93fb801f96df8d65d055b8b1431f23344aa075ad846418e9b, source_statement/4fa9d068da43c7dce82e8a323d9e20783f397c8ed7e9f4688baf5d51176392e0, statement_hypothesis/394738fae77760d2e7d47be10ecb61628c8652bd3752eb97d3a7f061f4cd1a51, statement_hypothesis/610949484c0543bca1ad54a9b7d7b94e0bdc8bcdd806f046afcb1e59b26e052c, statement_uses/2921cbe4f7ceff6614862dc7def78a7fd7eeabceb832f7f2a071876625850ce2, statement_uses/29e0661e427aeb8a41a4b989bf28a87c1a0fd9207fd047ac06f9ea1363f9d62c, statement_uses/394c2c5ca436393fcb3b08fd999f7a2010474876bc49b8257417102ee3f7e03e, statement_uses/c439cd99e2045cff96282a6736a8901573750706e146cdbdc6b177a03b355fdf, statement_uses/c95c6c07cd4c90a607033aeea234477700b3d4c073b4f4ce00df509acafa7c65, statement_uses/e665513efba1245f50a8b20e73b79a916f5123d1d163c732634a3afb6947e0a6

en · unofficial · sha256:d1b8e48f1efc0a9091ac9cc1a3b5bf61041b46f131d1f9c67746d52c3dcd489f

wps $f wff ps $.

Referencing norms: source_statement/6ff9f82788b9bd77d78984ba70659118e2a397ffb4684ebe8876cb89a927c657

en · unofficial · sha256:f0b9e9cefb1bde5f7ea43b32bb94ff641433c929109eebace94703e36417c1ac

a1bi $p |- ( ps <-> ( ph -> ps ) ) $= ( wi wb biimt ax-mp ) ABABDECABFG $.

Referencing norms: proof_reference/96f56c8df023130268521e02362ea3decd22df5ffa80014fc397b40bc86ca571, proof_reference/a9479cd60dc4213013b3ecf7909329709b063e1b57e8cdd29e47efd02c6506e3, proof_reference/befe946c2cbdbdae280897c250d76607762f275a957a62cc1ce03b24eb61dc2b, proof_reference/c1c0311c9652696f3c7637b8701f74694a40307be1a73ced523291583e812c4c, source_statement/80454402e5c1cc4fcf483ad197ce6e674d4d9f965ce8c09bfe3fef939ddc974d, statement_hypothesis/15738179f554c320544f7adb507f7065e51f90d75a863edb1d8175563fb85cd7, statement_uses/1cf44d190f993af5e1052a4da678c54e065babd7296bcd1bd37748504de61802, statement_uses/365e3940c80fc0d619b87d45b79ae6507df2f25387fbdb78cb80cb54480885b6, statement_uses/4b01b9fd812e705838877b0a4bc1f4d622ef475048000e74bd673da2d763db27, statement_uses/6c8c97aeb4bbd98a83b46837ff9a419d7b60b4195bc1d3f07eaa7f052b5034d8, statement_uses/8a7127539880dfdf894952dc8696d3218751a09c3e0267e5d522ec830ec7a4d1, statement_uses/e63d7844d93cacbef3a21be605f92f40872b3d8376e03b2ef244e799ac2d0bf2

en · unofficial · sha256:68d668beba6738f7721889273f1a202ac8dcb432f8f9c02497a183464f5eb419

$( ############################################################################### ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY ############################################################################### In this section we add the Axiom of Choice ~ ax-ac , as well as weaker forms such as the axiom of countable choice ~ ax-cc and dependent choice ~ ax-dc . We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead. The combination of the Zermelo-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satisfy intuitionistic logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms. $)

Referencing norms: section_title/1bcfdff21d92dd72eb0dbc1a4e54aca8a72125edf9a0a0e469923c8701449c15

en · unofficial · sha256:d903555e3d907cb1820982144b1ccf04a8a4aa554b5e6b211eb132244f793e07

pm2.65i.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/70647d3a384a8c599bbbd9e4a5f4e366e081433ebbdfbcd07b248937879b8e6f

en · unofficial · sha256:2d4c6ede7ff9b6076f473c78a38693312711e32d9c2f9681ea38243cab8102bd

chvarvv $p |- ps $= ( spvv mpg ) ABCABCDEGFH $.

Referencing norms: proof_reference/b09f6921892853ef6ed6b00c7c42d7d2e0f74db9e36acfd5556d8b4e09d0db48, proof_reference/d599f35fc5aef900581a98a870d28f1fa30363fb477163f6010b2e6dd8218bdb, source_statement/01fa8ee204e6d3ae33d2897f2d5287504cc25ce6e1b92d996c65b7380b613935, statement_distinct/21950c4e724fb2ac23df361c8e7a130a9e1186a3518904c70daf1555f031f939, statement_distinct/a9231d8658f955ddcc53c97cde1ff34688d6dd6a428c70d69c6c65fbcffa80ea, statement_hypothesis/4fc6ea526b46baeeb0a811c478c55a9ac40e4632533ea42f3e630a8a3014d2b4, statement_hypothesis/96060c8b54466ff1f54d4b8fc206cf13108fa55de6b84e39dd1d8a2521f81c6c, statement_uses/dbf14b1bf9a397c4044c5944a4ca6dd56ac80c25f25ce471e0dd0cfc203cb697

en · unofficial · sha256:78b58d31b9e38d40d5b69b4c0d547fea85f38cdac95b1e2bb4c55b62379a562b

$( An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.) $)

Referencing norms: statement_comment/3a48f4b6f769df89fcbacdd613c2d678da2d0344cccdeb350d0289f6494718ea

en · unofficial · sha256:6ca2dbce1fed9c59e078c42192a309a0fc57d6a39857b192f3e36db555e1d14f

$( Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.) $)

Referencing norms: statement_comment/434492b734f1ff6b3b70f38bdce41d9b1bf1e12fe9690e85b86dd9a562601f17

en · unofficial · sha256:b94633376000a530066b0b5cae337e7e82097ae35ede57117ea1d67f64e72aae

19.9v $p |- ( E. x ph <-> ph ) $= ( wex ax5e 19.8v impbii ) ABCAABDABEF $.

Referencing norms: proof_reference/1d0919dd10802e5341778ad3032425d72eeb46eb33d8cf2f01233367170ea405, proof_reference/2bc83d37d27eae95086f890f7d5831921ddbb46669f9c08aac295beae1b2a63a, proof_reference/7ff41da2d1cd1385d18e877332095d903d736ae6bf6db2c05a8a664b563b0da5, proof_reference/9930487252a9cba56342106ff9b8423e928605351a32ae481b568da198bedeb2, source_statement/309909d8bc070bcec2ec7935b217f52e094c5805884c1aca86ec87822302e901, statement_distinct/6b3428c827201c118706069f1f7479459b28682f120c1a8c27eb03989b3c2a56, statement_uses/0910cd6919e39533b6da78a9afdceaecf490b0f3d125face2afb279a0efe09ea, statement_uses/840925530368c9ff4b8a538172626b5f396a40c163b07e15033539f4860a13df, statement_uses/a9437a391e6ceec504393b34aeff463bf3246e8d439bdf057536fe338bb0cccb, statement_uses/b94215e70ffd6062a90eb7b71ae4aa21fc6a29b44801382bcaaccd4a142d55c6, statement_uses/c28947cf1d052e1e70781428acf486c05c62a62875f67a87a5b0cf06373f6c3e, statement_uses/ff18fc5500ea291809aa56e2c1ec64c13b8c73e68bfa2eb86b6a9dc6a106b0b0

en · unofficial · sha256:6c529c8084ed1ea7b8df51bd045e727874a5cb4c07da195f003b5962e0c86084

$( Infer an implication from a logical equivalence. Inference associated with ~ biimp . (Contributed by NM, 29-Dec-1992.) $)

Referencing norms: statement_comment/6e0f457294ca82257df8de613317ccbed9db36a8b4e4355ced4f5c3f6da00401

en · unofficial · sha256:ce74f0364f3baabe255cf830e4a454a6afbd41d5d087f26fa7d0ee701eb30d32

mpbiri $p |- ( ph -> ps ) $= ( a1i mpbird ) ABCCADFEG $.

Referencing norms: proof_reference/6b862d21c176bfaf791c022babfa6e2bd8dc03fb1a0bc6922520736ad6810f41, proof_reference/bf46895053acc27bbd2d7d509c355ed462a3e30c62e1246ab018c4b9c8d3a184, source_statement/b47a1125876ef3d64aff2ec905e79ba1e2db3e15de05033e894cf6f70ee88ab4, statement_hypothesis/315374ccf2d60a9611eccc4e3693636ee31e8083fdf61829876cc99cce959f8a, statement_hypothesis/56f4b8c839655d49cb3239479441385329f0a7c3362a281bbe817e26dc94305b, statement_uses/2c1f568161fe6a54af497d61ab6aba812aec9222741f7a0a959254b5a6e94aa7, statement_uses/79db5300df67bbcab7513b1904db7b5f2e30c426697927445dcbd081d34e587c, statement_uses/bde6cb4efdf110b8d3c65f145634ddc865954b27b0955c61241996aa50fd2eec, statement_uses/d4f709743944f7a0d0ffd8360249a82ab2fa4c71595e5037b78453c414055ba3, statement_uses/eaac89bc10b4104772dba2c5544665f848542721c4b913f2182958931bf65e9c

en · unofficial · sha256:2db8991ce9e456c3b8c3025d9316f3ed540bc5de0bc6b5faa146ba6ad09419db

cbvsbv $p |- ( [ z / x ] ph <-> [ z / y ] ps ) $= ( wsb sbco2vv sbievw sbbii bitr3i ) ACEGACDGZDEGBDEGACEDHLBDEABCDFIJK $.

Referencing norms: proof_reference/12b5ae88608dfd2b588e29af8eaa39bc48567e98000b4c162dd901a02062f99e, proof_reference/3f028a33134c1055dcb5a8d86b5645f4ca05f0e4eeebe83476246fef0ed0cf4e, proof_reference/409b083e77054a04f86e453a02d1cfb7b9237c897ed364c14f5d6b79270ece19, proof_reference/bb4e6fd93463e7ab55f50c1c594cf5fa2dc7dc5455ac1cb15bc6eb1ca0cb8c9a, proof_reference/fad3f5abf67f99169dc87d51ed052bbeb10267e4d39ff91462bea3eba446f413, source_statement/4c1608f6bff2cb7e39f7017932ee72d06600330a9c093bec07bd43b2ff233c37, statement_distinct/19137715bc3011da83d74b1a2c9556d1d94f46a132a557399295a63f0acc7828, statement_distinct/569c9fef5a16728ea2e4856a50f57a8e2ef756cf6482966ba75d13a142b47213, statement_distinct/b4fe8e639f737576247a2aacfce836f99c68b7a21cd32d22ffd6a3bb09a87975, statement_hypothesis/43d84b82c273235754529feb90b8960357cfb7f72edc3ebc385a9625d877acfa, statement_uses/0bce5c7c90ae2f1457f703e819484f270c73b56312eba6bcd9e5534a6e8b8461, statement_uses/779b47069da6dc6172f2427f3ce8fa9eb322f398e7e192ae9d8a3f685924b6d9, statement_uses/842c5eaf93084b9b0c9f0969fd04304e6afb5c7099356848791464c923f76cbc, statement_uses/8be3e8eed7a7ea7f1aacaca39d7338fa997c68b0c4ac790a3a27686c9c5d4712, statement_uses/a635b1318ddbfe33ac4c7d84c2729ac6baa11b35666b1b03ed383e3449cf664c, statement_uses/aa64462f8561baafab00e11bd46b8be23acfafc9c1245c77c5ca28580dc322c2, statement_uses/b02776ea0a186d8e4c743f005e6a849148c6d2fb169150e71d66db3534a3bc60, statement_uses/d1c6b78b20d164584de0fb5dea51a5ede6848361206f86f1e4fb85f9929f5a9c, statement_uses/dcd6256b8dc91e310c3402d73acbf93fe69c40590fd819eb5e972bb734d88eba, statement_uses/e23302f7fbf4e0e070d57b8a7901d3282118eb85c821667a4688293fa4d0e01c, statement_uses/e6cd83352aa52a974e95a8ed5e39b59294aa5c82220292bc3693328445d96ffc

en · unofficial · sha256:4b34c46ce204b23a4d5ad75dc2606c942a0e6c1f0c79221290cc4eac8cc9865d

$( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Restricted class abstraction -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $)

Referencing norms: section_parent/ea434a5b55263a6afe2bafe66d72630499f4167c8c7d7ae98ea7ce36bd1379fb, section_title/a3a0c074efed923a01cb99c8a88cce91b848f8fffd07b1280490fdf4f3b82490

en · unofficial · sha256:0f640eee56a5eaeca05abd9779488f75c70daa8ba2155a49cc851220c011f566

jcn $p |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) ) $= ( wi pm2.27 con3d ) AABCBABDE $.

Referencing norms: proof_reference/6a45fe9179c9215693a98b4f212b1885ad78afa4cb4eda806807441112ffdd04, proof_reference/f3e599e92d2b51dde80eba867827aac9664d38232c4b4de551e6108d6f4e7d19, proof_reference/f566675a1a02e2227eee8693556eddfe7bd59e76f50c424def021f9240953f1c, source_statement/697b013a52b323fe5c3de53485fee34568d9af43713b56a141d44167469bcc33, statement_uses/0604682d00a575e813511f96918613d5fcae7267c31b18227bd8ff05ac113407, statement_uses/185ab6a245a64d4402cd6edf896b67c9481e440343c8ec14b1e582361157357c, statement_uses/3b727b62b4718704b38acb9eba32128af4142e441c59cc0e5a28c2618dcbc7cb, statement_uses/ab6d827895f57ff66466d4caa63add90a249ba1431e472e2f6457f36018d5bd6, statement_uses/ae070096cbbd3ae4ed6dc9215f9d50b89035e1eea9e81648ef38ae542d300054, statement_uses/dbb40a5508c7e585d014c5218ff43adb35b5887ca52fabcc75c9b3a9beb413c8

en · unofficial · sha256:e7846f6e2e70c7643ed4077c7e80d809b3942886eeaa0b648ae5050e36b88593

sylan9bbr.2 $e |- ( th -> ( ch <-> ta ) ) $.

Referencing norms: source_statement/ab6e31703766dda6d7db8258978478104ac5f688d654af6a468fecf0176f8cb3

en · unofficial · sha256:e9e6c0e74af502bf9078f63d467126598f162e644a8087e14a18e15e6adf87b2

$( Introduce conjunct to both sides of an implication. (Contributed by NM, 3-Jan-1993.) $)

Referencing norms: statement_comment/012fadf34c78d021aecf69f2e706d3114dd6c301f21638c40c4c269805fb4be1

en · unofficial · sha256:fa7aa250f96b52f195680de7fd17c4138000b63985e686ce44c058e791d283b3

alexbii.1 $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/a08bd5c1dc389810736e8502287212127829b338297c4c227502f678b0fcaaed

en · unofficial · sha256:91b3f158d01207a822d7a4902656d90e42c11a378d712da687cc2835e0ce5e52

eqeq2d $p |- ( ph -> ( C = A <-> C = B ) ) $= ( wceq eqeq1d eqcom 3bitr4g ) ABDFCDFDBFDCFABCDEGDBHDCHI $.

Referencing norms: proof_reference/5402e641250a937203e3609dbb5331268105a1cecf403a6bf5ec8b08f116ad07, proof_reference/637d4afbe0ca87cc9607c4b8441a145565e5a97946d24901f700735897bd9cc6, proof_reference/f2bc26f69735599baad82a3f2d8968ae12bc86e92cc2ca7203c3191a574b2319, proof_reference/f394524b92bc8d53aaf3cfb979803a7083cf2813350d69712339a9f8ca3acb2a, source_statement/c7eef31fa57b07b44bb7206600873b2b021a7e2ac3cb6da3f7e0255d89a6fef5, statement_hypothesis/5a6eac640057f760d57260e30011d737c77f56f225eafb219d3d013ddf96dbf9, statement_uses/0c288ab9f85f4e5bfcb81220b3e70b4576f6f401215b78654a41da282045bf94, statement_uses/18ad665abcfee77b0676dfb70856319aad6b75e350d6dc54103457bb7d9fc227, statement_uses/6876aa606629d98ba8f7051e76d38c08368184c26d796659421c4ca76ee963e4, statement_uses/739430c06b88740081e68158885a229f113ffa8ed0a4da50c9efaae463045f84, statement_uses/7542327bc104456daf79f11c0e0cc1c42168cf70c9df2b333bbb001b9d072aa5, statement_uses/864a81a3612b48bd8182b1cebfe9790629eeefa59949235ba7f60308aaaf593a, statement_uses/8bdd4e7635e7ef08fa4f328ce9ea78a2ccb49601516009e2f5ff84643a8d56dd, statement_uses/a16ae6fa3103c8e3d21dad763ef11f2fa26e21a3dcebd3903d93b59d855c1e67, statement_uses/ee1d02a8eea2983757b9170e949c8e37c61da83bbdb0c20f756120d3231f95fc

en · unofficial · sha256:93800e12a2bbc1fcbc664cf72941e2198d4e967feff99dda76b6a0f2d0f1a524

wa $a wff ( ph /\ ps ) $.

Referencing norms: source_statement/b368cf4c4740062fbd6a330403146cb19b1dfcd00bd1227f23f81aa43be0e8c8, statement_uses/0c3f5ab4512ce9bb0282d45a8c1ac2b6705ee21405faa067df35a360c09ee600, statement_uses/725012578a4f42e24b8e981c83265cbe03ecbded76be3c1081e6843351913015, statement_uses/bf01bda97efcdb36a27bbe3416e6d7723a81e8d81c294bc846104566e817912a, statement_uses/c678ba1f8d7eb580d3be7039a7cde71aee285b4aeca542198ae9889f37e872ad, statement_uses/d2a42592cde74bf5cf53b9d3d53a4384f79c79107ab603366a1ce5c8ac5c695e, symbol_name/e027e229bd0750680cb83233824db52fe3fbc18dc75575a3daf0331d0df70722

en · unofficial · sha256:7c0d3ca65ca817aa5c7d240c5977d6886845263193a1a11acb45e8888de246ce

$( Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) $)

Referencing norms: statement_comment/fc0e1aaf4059226d674de8daf79624a2b33e39910e66ea387896f4d9b7536485

en · unofficial · sha256:989bd095700b72eb245810ae7a1e7ec7fd987fa8d6958ee969c4ba63cc1f9d54

$( Specialization, using implicit substitution. Version of ~ spim with a disjoint variable condition, which does not require ~ ax-13 . See ~ spimvw for a version with two disjoint variable conditions, requiring fewer axioms, and ~ spimv for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.) $)

Referencing norms: statement_comment/1cb5a89fffef5f1386164073b5f187467f9b300701ceb4e64e1f52fd41f61fc9

en · unofficial · sha256:7c6af45419c38c5cdfc9b241438de9f994ff91828c866a7be9fe49915bdaa79c

sylibd.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/a81f7dec87d822e73bcda1909e2ab420a681f9cb76d3b5e73eb00bcd38116f1f

en · unofficial · sha256:ce0076315f4461ed57b26d5b7b6674b70c621113e04b7056f02313efdb1ca66a

sbequ $p |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $= ( vu weq wi wal wsb equequ2 imbi1d albidv dfsb 3bitr4g ) BCFZEBFZDEFAGDHZ GZEHECFZQGZEHADBIADCIORTEOPSQBCEJKLADEBMADECMN $.

Referencing norms: proof_reference/11111e3d438e0186bc0dcce61c3d0dd1621705e9246b3a4a6b4de80353dfba10, proof_reference/25f6624acb981212292dea606da0bd3108409a48a8d77d38cb38dd1c26b62120, proof_reference/32b55b719e88c035c58b45e4b903fcb1b9ae8b719fd0c84b7bf5e3248573b683, proof_reference/377207acbfaf5d2a7cc36db8f6056750d4a8128a64469751b99ba0d9f2be5d5a, proof_reference/3c71fe5ae70412df77e6aa87e23a2044d29110360dfe18f40b6c62d04696e623, proof_reference/6b83c01bd6cf4d63533116a01a6b035d91d15ec83f7d49eeaf1e9fa748548e5b, proof_reference/8730c1c210913ed3433e9f668ae170386b9f0140c86db682c8e5c369ebbf994b, proof_reference/8f7a94f0e4de37d4bfa67f5fd4927d86d49765a2b621ea2edccee4bf91d7503a, proof_reference/ad4b718b75fce45a2e5da4b7a04817756eb54d009808d414c811f7ed476a5ae3, proof_reference/cab001755f5d26430c91d7091236a1743efde9b8014000876296cfdf4304d17f, source_statement/08e4a7223240283f1d0ec06fa2a79acf1af5812a01f5a72273da9f9fcc1644ff, statement_uses/004feb029caaa946287c3ac52ee0ea02c5dc2a562881a996f0d1d6fcb19bc32e, statement_uses/3fb9aa38e841d5f99e1fb950d6aa84d61f8b2ec92c60efdd2773f6fcbb612aff, statement_uses/4ca4e403f4658c62e2189a3214909348931a586a155c7bc420da40d0ea1817a2, statement_uses/7ee1c7f324ec3dbaa6d3dcdea6a79e801b95d437626ae99b9d7cc1675865e374, statement_uses/819a1c142135861705d4813000c481c9edec1fa38b76a6a57e89ea7d2544c628, statement_uses/8d0e832f5ea69894e9b548cabab2ceaff901171515d00ff3b34f98e04595fad5, statement_uses/b001d2121cb5f739344eed9564967e15a65f6fbc8e38dc9a99e0a17f395e8a3f, statement_uses/c2e1db00ec2dc0145e5cbfb5de56f65858cd890add2704dad164debdcdf71bfc, statement_uses/c698a337979c53beeadc4d55440faecd34b5b41aefe2c4efe3e95430d82b245c, statement_uses/d4f59e9c30c5ace05ec4a9294401447b9a0e1f39a231afb4271d79ba5edee0ff, statement_uses/fae297d57ce98ee558037ae08edded17e14dd0ec7b4de10b6da08970232f5911, statement_uses/ff28a4fb25cd2485b6607668fb81016adb617709f72183ef0074685aee962600

en · unofficial · sha256:e1ce7928d489171cdb4ee324985659d5d0cc7a6ba7ef39f99d034bc1cfb653f6

con4bid $p |- ( ph -> ( ps <-> ch ) ) $= ( wn biimprd con4d biimpd impcon4bid ) ABCACBABEZCEZDFGAJKDHI $.

Referencing norms: proof_reference/639f95ce96caddd1ff02142fa3dc70307cf67ad1236ff4de44b303ec50c6e89a, proof_reference/8689bd58b93cafb02c852233777935e5cc6a56aac75a6f05aa1e752249a84a78, proof_reference/8dd196bf106348eed2c5e01068735a0ff2e6f1fd6765ffc5314e378d36a91e70, proof_reference/a504b008e02b6756502436dee5817b4b16a9ce2a6c49601f46248715f7e2f03f, proof_reference/c27b344090ff571b599332365a3f05c9422b0719406ea36f226021534b5ab9c6, source_statement/0ff858f75477769bc1d59a5d3f31805e0e708ef31c7d139b2f39286a332b7cca, statement_hypothesis/dff90994ca4d92a57a70924dfb0672e4ea276149abdef8da3f34f4abc57e84b8, statement_uses/183b22d42f40b6929105792ca7f33939d9e7bc6db64457dd9386ccd18e863626, statement_uses/6016ef6d26e480e1f664f3d933eac14487cbd180e2d610e2340ae8dfd1ac1a2a, statement_uses/6c57c77badf41b85d7e01a29f7f44b60f9e1d0c6c41cd6fffd6074cebb8a2698, statement_uses/780b0e1dbc8282b3e3152c9976c953cfd43d15231f88d9bf21ccec27eaebbf03, statement_uses/d05a4aa97d4c42bb6984f591cf68778865f764f86775843e3e4718d1f2a7b120, statement_uses/d0aa7ec5acc0b867d4d16e08a1346a05813b3e96f073c36c339cae3051a1fbcc, statement_uses/e5d21b01f93a154e0bb7956c0df695d9dfc6fe9b47f1e127908a6c6d21f4b2ee

en · unofficial · sha256:ebae5784aadaa7ae8e5b207a5685f1d16efaf089ed78a6b32f37340ff05dcd6f

$( Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) Adapt to new definition. (Revised by BJ, 1-Oct-2022.) $)

Referencing norms: statement_comment/cac93c6baeb374dd0001c076196c47722b3586e772ec891f794f99586cfdcdf7

en · unofficial · sha256:927975753a2d584ceea590bf1fce9da2863ec8dc6905690163ef605bc9fbd5cc

$( An element of a class exists. Version of ~ elisset with a disjoint variable condition on ` V , x ` , avoiding ~ df-clab . Prefer its use over ~ elisset when sufficient (for instance in usages where ` x ` is a dummy variable). (Contributed by BJ, 14-Sep-2019.) $)

Referencing norms: statement_comment/c3b80b86313a8cdf9ba8096f9f5684646ad1a625454fbc4c9e6727c867da0308

en · unofficial · sha256:d564a351f077b9875939df07a15b83404861f770a2494be6daecb8343b5d40a8

ax-rep $a |- ( A. w E. y A. z ( A. y ph -> z = y ) -> E. y A. z ( z e. y <-> E. w ( w e. x /\ A. y ph ) ) ) $.

Referencing norms: source_statement/961b0d95b95eafec66ff651e7256b3bd639bb823ef5f8f7831281f74ca7e68e9, statement_axiom/c0022dbcdd8ff77fcecf75e431df8904790922810a1dfeb2332d8ae1008b4faf, statement_distinct/120566340142c806395b8bb6c90ef19db238a99644f6fb88ffee69d6b1725408, statement_distinct/5a59fd8e7fe32d854c7a7d4d43c8bdf57ef4e622b8a1749f1111b372defb8d42, statement_distinct/8ad230345e0343b078728079e731bc4ca76ad041acebad1f924bc1835aebb823, statement_distinct/b968b6fa218847c3f352871ca8047f89229374ece26ecfdf8bb506a8daef795c, statement_distinct/d0289ecfa863e0d050d02d40abc98b517025b738c1e403761eab492680f06d9c, statement_distinct/f90bd9e6f1f380949cc0342eacdcefe772bd3c8cb7254e7f2d2f1b2b312728bb, statement_uses/20ed404d373b4e105597fd2f98740e547cc1350b54da068baac75df83b2c7e2f, statement_uses/21066080a00592562cdc37f7caa052b0a7a09b2c2517283be9e69667558a129f, statement_uses/305817263d5483b6c9d25b1135fc9449a85a2c0ff5df08d0ee61e6ec4bf8ab59, statement_uses/41f64f9173f491017b7c3d8a4dd711731130f895b6f434220e9d35637c9d671f, statement_uses/43c0a67a8a4377a891b34b8bd49c09f5d830ab0d787ffec99a6931acc3842c90, statement_uses/70f5ec5060a14fbadc54d49a644a6e2d1748dfe58270adb3392ede11886c38a5, statement_uses/7ffc874cbcf908831c24235887111a23674cdbf3416593f5b8f548147a3130ee, statement_uses/952686158bb61ccd0184ffa7872e05d0e1fc0e0d1756bbff90a16e01e0a643a0, statement_uses/9dabc593a4d1470340d3bde7081f95aba2273fc6be31146d16de04be14683045, statement_uses/be298318fbc886f613fc89c2086c2fed8165f70f472e4dbed1e8e05e942f549b, statement_uses/c1cca2e39b6619dc060bb33e0ee9747e9048fc536bae259cc9de8290e59bade7, statement_uses/c7c7c68c9fb7b149f57a879f10d8fa9988b3eb5ce2639e9730a35137387a5edd, statement_uses/ea8a402424bfd212d6ad65ff468f351b698878bd1b112556322ff42b3612b925, statement_uses/f28d57924cb57e7544b8672c2e900ebc2e50ccaa8f4d1f7d5c6d7d5affa19627, symbol_name/8fe0cea3f4e7665d412484ad931fd8ae4e95c31110a8660521522059b4dc9666

en · unofficial · sha256:0a588d871a5943d0443d0f1b4d38ccaefcb3786667f8838f5ccba38a923b3370

eqtr4i.2 $e |- C = B $.

Referencing norms: source_statement/34b37ac1d46bed12991a7f5976e414a59316ca9c93ca2b1f7bbf0ffb52aa43f5

en · unofficial · sha256:600602f0c713a9132bd2a6a35e5d26078e89af1128c9af906acebd20e0a9b8ea

eleqtri $p |- A e. C $= ( wcel eleq2i mpbi ) ABFACFDBCAEGH $.

Referencing norms: proof_reference/1f395a5fabea37363de741865ec462c4c67001a5f4dd1bcf466f75b5b68056c7, proof_reference/54998034f47f1fb605eef30f1ec86b43dcf376a74df64fa857a77fb8a57445bb, proof_reference/f4e226b9d6f288877a462468fb58ad1b0c600e7905fc423f4e92642b8955f384, source_statement/d5e5180421b58da56ed98a00f372dd8f30f3e21bd1f8bce2a6b1ab0cbc59a6ff, statement_hypothesis/468ac62ef15028e121cb7be5ca06d3237c10f1e7b8df8294ff2eef14e45d426e, statement_hypothesis/7ab0c2cad758f957a59d7b373609a6354d7b35ee7a881410d02db9bf0538ac63, statement_uses/3dfa18fad6c113218a433131394dff82892aa1173ba50e11a6d34a08e7611494, statement_uses/e65d6671821cf5adf1035feeb41a4dce0ebab0b92716000849a3bf0996afad01, statement_uses/fbacb7e528ff3f970e948ada5f575c0b035e96709022c4cedf61b8a916e7e47d

en · unofficial · sha256:73dcb467d44eb782b3a2b024cab8660c76681d42ea7bd5c13cd6f1af410d54a3

pm5.74d $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $= ( wb wi pm5.74 sylib ) ABCDFGBCGBDGFEBCDHI $.

Referencing norms: proof_reference/2f741e98d457f6709c2df7723c25dfbcafe04f8a60b532d83e7dd01ec8327389, proof_reference/46142b6358c77d8f6b1e73ebba8effdb06050e75232d574200bbb682c3427d04, proof_reference/865549cf55c90afc4ae69471428a3d379fb210480d26e5e16e2393d75981a7ec, proof_reference/9c1ba63e14cf2315b2bd41e5fdf380f1247eb3a425cb65f2e0f07095ab48e72a, source_statement/bf7a0c2430ce56d7f2e06a3166e21a4b66f8fa230bf06189b230ce041778c919, statement_hypothesis/ae51a3a47a5adf7c30ebbaa78998437404c1791f1503ae04b312c963e6927850, statement_uses/1fae6c9d9b1ef8be4ccca842aee8c6de7a100eefeb401bd749516b63bb2f2dd0, statement_uses/49795e873c7167ac494323ba545e0917b3d3f4758e343a550b5a214c2a068db8, statement_uses/4d725835fdb07c5685e50410c2d4708f005bf887d394bd27a639de1f04a4c57a, statement_uses/732f9671c89db1d4a2f2903f191f855c54a417b74f8107116c7ff2ce592cc711, statement_uses/8dca843c65438a80eee1c3e7b8eede166933a87cb252852649de7a0d37bea2bb, statement_uses/b02869dab15a64b62849140e2392c0358cb0c83a9ce2e99cff38f39930eda39b, statement_uses/c371f51268c018741a060c5aefd3fca474c9f33caba6266534dba3fc8f52c0f0, statement_uses/d3236455a733bcce662ee6c43cedbaeb09b1e70d3e371ea30401c45226a024ae

en · unofficial · sha256:0d250670de2386c5ac19b1e5939b172460e8aff5b9a9ca194b3ecd8ddc508195

alimi $p |- ( A. x ph -> A. x ps ) $= ( wi wal alim mpg ) ABEACFBCFECABCGDH $.

Referencing norms: proof_reference/06b622100a9fdc8df2229565b3bd7c909d12db2db4b8b0319b3648e3539af97c, proof_reference/c730fe111444fc69813df8ba59ba485841b476d494a146143bd33015ff79ca0f, proof_reference/d68913c97775c22488cd3e8d47454519d9048f1e66307d6bf2cdda87e6fca872, proof_reference/d77272ce45c7823419c2d20ea2c646144b8dc4db1fd997bc6b7802efd82db1d4, source_statement/7c19bee3ef140757377efc09cdcb62c9356efc88b6e98b3fe7c6128312602cd0, statement_hypothesis/361940ccd3a93657b900ea2c30eed20f1b4b658516b8b47a51fe5f146cc216ff, statement_uses/3cd237ae6da1422822287adb8f27f7138b733dd3051133b9b838e1385c1e2294, statement_uses/47b0eca95521f964c2c5b27ac8dc6423ae69a70dad4b882430f6e404db6bbd02, statement_uses/56f39ac6c1edc94c4579eff7204813f2134fe18d14efba59d83a9a2dcded20eb, statement_uses/63f9024ba8336bed9f9e27e126f5e7b167e5bbfd12fd81c68e53da8f854ef1d6, statement_uses/78618c9e7649ce28cbc8a980e072908b127257b290c27f26cb0d84e401ad8a9e, statement_uses/d634aa72fc3f47129408f3cd56840c5f1e07d950e0f2bb8f270553b408285667, statement_uses/ef816cdef1778c4b7921808036398930b37911174b3085e76fd702862c407a85

en · unofficial · sha256:734b6e8886390c463e555a933ae16228fa9f0384ae699358aa776a21a3f9ac68

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Introduce the Axiom of Countable Choice =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/ae4ebbfad64f154d3a691919d79d5365ca06e7589be71edc7197abfaab49e8c8, section_title/5546cb41d00e6648f41cf0e81cea18d4559a4f8397f3d91283ed5b7d2d165b93

en · unofficial · sha256:5c3579d23bcca3e7defa4ab3844ebc0f6419df3782b61049f722e6ff943336d5

impi.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/21f6388efa6ea3be451932938b762572b56c27a24d0018c765853b93d70d8793

en · unofficial · sha256:49df8a0061a7520e03cb8835405638ac118dc44854d2b55bd340e2c2eaee3568

biantru $p |- ( ps <-> ( ps /\ ph ) ) $= ( wa wb iba ax-mp ) ABBADECABFG $.

Referencing norms: proof_reference/298170c9b91d6229cb1119c67e57d3902285385be2720e28bb9d1dad9d33f283, proof_reference/6362ef8d3dd95c4ff5087bef429447e4f0d2ef8c16dc8b150c3e4f78093c176d, proof_reference/b877203e289edb6a6da9cc98127866041ce08bb789e2f977a6bf3788100c2542, proof_reference/dcc5d24f9e21f8e42294bcb83d5346da9284f371505ff31d1157784df7ed6fd4, source_statement/07480215b5e4aa4aa9205e99a3d905c0c2594baf75a6dd8033979084ded3b3ff, statement_hypothesis/f544d47c42e8b06dd53b3f64bfc7820c827010f22261dde2952f9616cfee1a1e, statement_uses/4bd77df4cb53c3e8adb4c5c913a3dd738764b5e409ed42d7d379ecb136ad2b8c, statement_uses/75a78e45b09ba6d38e4ff08c8c24f631c6980522869321644f0790c23047ceba, statement_uses/778bd6adf24bbcd0700c2c27036489626151939d6e3fe01c41d5d7a337d0bcc4, statement_uses/7b529104a4ba301c622a9263603b50aa10ae7ecfb80fefc3deb6f278b610671c, statement_uses/bead4fe67d78ed9077d78d54b0fb7b73b450e451cc3331333d7663f1728dc819, statement_uses/e8775ed9165f991b25ecaf864793b3b3a16e29d3fb3e96e6e98fa35f79201384

en · unofficial · sha256:d1741507447d00f9218cf03e7a6281bc55dc0c3d43d14140e53e3fa307a3ce78

bitr3id.2 $e |- ( ch -> ( ps <-> th ) ) $.

Referencing norms: source_statement/65f9593b93b516ba0e3ad330cdb32f1a04b9b48b014fa33ed90bc875c1f95672

en · unofficial · sha256:806d55ada25a18a635a20bf6dc9bd31a3468a5a1420f56416f8c6776ee14e8a4

al2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.

Referencing norms: source_statement/588350f9048fd5798e4c7b5253a2449c6cbeca6bd00aab2167fcb1b0919a8d53

en · unofficial · sha256:2405782bf40e701aa935ad0396fdbd82f058ff07e51150b31bf47d0afd7434b1

$( A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) $)

Referencing norms: statement_comment/191e09002f9305aeb299b1091cd35ab5b3c4f7baa7a42544a0d7152d88f8941e

en · unofficial · sha256:a890612607014f5ab4788f9f17e3f22e24074cc01e7acfd9f4f362cacc088a08

$( A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.) $)

Referencing norms: statement_comment/68bec2f5967f41efcd23db5ff28a1f08bdfc411f46924a306bb4eb34c2db245b

en · unofficial · sha256:78889a5bc6e95640ec103f1cabda5ad3696c699efa91fca0bdfb18cf39ccf57b

$( Inference from equality to equivalence of membership. (Contributed by NM, 26-May-1993.) $)

Referencing norms: statement_comment/0c3880d0050e45b4eea3c66cdd323023f36f6290a3358c2636142b8cd6ad6e28

en · unofficial · sha256:714e7577c71141ce00019c3f791dd7ed922263ca5c190d80553ab823057269cc

$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)

Referencing norms: statement_comment/2a48031fc9d7a786c85905109a4ceddd978256b699a394225c89d0c6831ef18b

en · unofficial · sha256:8f7ed7d878637875f4dc08932237aa2154ba6ac0007bc8ced7f4cab817254023

2th.2 $e |- ps $.

Referencing norms: source_statement/544f749f7536a31ecceb5e3d9270eb88ceb7e1c700662a6bb3037feb1806e3e8

en · unofficial · sha256:3adc6b183629727b58a75452b46b98c493db038858e59f92cd54bbb9b5f2261e

dfmo $p |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $= ( vz mojust df-mo ) ABCDABCDEF $.

Referencing norms: proof_reference/2588d61c45983b17aaae097eb85a98b3252cf9ecf9917106a01a69fe08a8d360, proof_reference/7c32de430364022a6eac1c4f19e4f352c7fc50e27b27e386c69850c84473e7df, proof_reference/7e52eb37e3d3ce65e13717d0bcf0fe55c83cf1a9a88717f7862f86c25c01835a, source_statement/523965dc45c76b32951b34b43393c46ce887c1f98649468cb3289f2b2a353c0b, statement_distinct/2a0339e0d46c51de81f7c8e0af77261a01a1e385f55b4b712350df6154add211, statement_distinct/c000fac4f8d187ff19d71e9f4c522379a32ee9cc308676c17040a31e5854f3a3, statement_uses/37dc5d4aa94ac65b19b70cde9607f4331766bee2c0f3f861902e495a61726a0f, statement_uses/42e5ab971162596d3b70d50caae121a925381a18cb84d2ddee53fb73742b4650, statement_uses/4e29ee259a7ded5ea5797f8f70dc95fd5a3e1b78e3d369fd781a894db1cd55f0, statement_uses/5107dd6eee43f5de4c0110e794bc2c7307d2419b41196156fcdcd667007ca9da, statement_uses/6e67ad37c61aaf9551a939724625541a61a5d6cf11b0717cacc9f7c4de502bed, statement_uses/7037f866bb363b6db324429e9a2ce4262222b13ddca005a392a0cc645a2649ac, statement_uses/86e0fee3fda461342394662d2e7a5e4a01f31637ebadfb0314a9b0cf2ebfcdcc, statement_uses/a391bdbb74816b1d877e643cbd2b4b3a40bde48af17f75a78b23a6561ad70bdc, statement_uses/ae25256674be6da5ab41a79494728dd58be0f3a16c52ed5e088a9ca771386c64, statement_uses/b8594d2185a3a9fe11274e8e53bd92377d14d324ecdd74d32ac744028dd43743, statement_uses/f4a1f3c279c8e9d24fbd1eb3f74c287036b7688b4e801ee11284dda0fe144809

en · unofficial · sha256:2400ca578af4b0612c6e9834a40533a28c46dc3cf12c4bdb9648be04ae67cf40

nsyl4.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/b60da80ffedc25f2848e9bcd54f3a23e4cae8ace6e1d195187b60174aed2b9d1

en · unofficial · sha256:eea2aea1c297cfcba4968614d7b45bcc5c8f5815c6294c78b44816e7ed77d4ae

bitr4id $p |- ( ph -> ( ps <-> th ) ) $= ( bicomi bitr2di ) ADCBFBCEGH $.

Referencing norms: proof_reference/463a4085d3440f7d18f09ea72ccda3c63269da7f75b2a5de4883edd03d397266, proof_reference/576031ab4201bfab652964db98436e6a92f6ef61a3dd301bd7d821e316e1c3da, source_statement/922dd20868c151933553067a84cd96fd8b79a1b7d4c00cd0d01692ad7ccac28f, statement_hypothesis/48dc6a5550dfbb7414213e6dd86dbe241dfc4ddde5c6d7616f4e56a6d223c0aa, statement_hypothesis/90648c0ceb2dec5178a70fce0899de6cb8ccd294e0abe4b8aefe9ed0b0ce317d, statement_uses/104710cd61020cb510444a29066327ee6d133adc0562f11fe175e178df6efa8a, statement_uses/17486f433e389fffa2a98cd322c083fbbf9b24b8a1d200695d9f12b1c4173db5, statement_uses/1a7f4944d53f11fc09e2690c522d9a867901ea20384e92a5237e7cfbf5530b84, statement_uses/3ca113fcfbccf7b441ddfbc5c5c28774e4bfb777bbee31573663acfb4c15f294, statement_uses/5175d4a384e09423e5ef336610a65e7c3676b517e0594d40ff50af472cc97e79, statement_uses/78e3e88df7843b529c4d55fcfbb325f1765807afd501ef6e1a70aa10d1ba7623, statement_uses/f48137689200a18d381d9481582d527a1e1237bdd0c160a9227f3609b746ad5b

en · unofficial · sha256:24f7bfff68ea647a617eb1def081495613a57f66482553ddf15edb400406d834

sylbir $p |- ( ph -> ch ) $= ( biimpri syl ) ABCBADFEG $.

Referencing norms: proof_reference/1f4df7821e6ea9ceaa8b34291f90a17e827d40e277b98a2874163d649c738369, proof_reference/caedc06f1095d5c00ac75b59aca03f77e519a9ac87262a899ae174d27a8890ac, source_statement/402e85bb244260a94e9d0be7f2a970fa3d77aa8b8fcc47983f4b0e465476369b, statement_hypothesis/45f09f3521998a208fb37d9b699964ca7e79ebd079b035c54e8721ed374902f8, statement_hypothesis/4e0458f3f5fd574a5a98f7b65ecb76e662b685ff49031f8b7af8929730e2cc1b, statement_uses/4e17668ef9236b799a8a02d27e44e462edae1d72c4a06f94d210a3c7c9449670, statement_uses/6038041e2cfda0f0c17d7aa2418edff030556d994e5d153871903aaefaaac543, statement_uses/885fdccf8ff8da8f2deae848e45916790c89790a0299738fff65c4f7862b791e, statement_uses/928f1cb5d93be9b5c365a3ad4776b5654c34f506459296e0395005bf5e042f66, statement_uses/c96442bc857c26dd1a27567b7f94f437c9c7aa72c7e3866b03d806efddaf790a

en · unofficial · sha256:d3d9312bdaab3cc759b895f6a79296b4cb584ffdaf22d7d7190767ed3f9ba3b8

df-ex $a |- ( E. x ph <-> -. A. x -. ph ) $.

Referencing norms: source_statement/d51ff1dd1f89c481bea17272c448030bd005f541487596c9e530bb3314131d3d, statement_uses/3cbf9d77eedf1ad23c6c37ce3646381fdf98c2b49046465da1710d76244f4cff, statement_uses/4b9e435553b9112dbac76cf1074eff0b5bc2adde7f34c43ec629e8ed953430de, statement_uses/4c67a7a19f03360ffed0a1aae57c332a59b3fd0fcfaf50c957dd5d31b6b7f205, statement_uses/5a82546a92f8ed8dfc24f57748d00bac9d2c829652ecf7bd1691c4f200119fd0, statement_uses/6033fa3e4333b793ea1779858e241e268a90d458a00c5f8e8546a06dc412bff5, statement_uses/691cdd7fd08909f83e5ab585f49644f32a081143761b6466d83fa13208861904, statement_uses/7e2c6e6c3d3715a633f26cb14f7b43680e87e703af7807b25b2eca1214ab5f30, statement_uses/8be3a827cf0fd817f211365c26bc8647c03e78e74f0dc3423a32e5564f3c786a

en · unofficial · sha256:a84cbd159854693a73a2b286954d504ac6a356f9798e8b23fb2d075a6dbe8fa8

alrimdv $p |- ( ph -> ( ps -> A. x ch ) ) $= ( ax-5 alrimdh ) ABCDADFBDFEG $.

Referencing norms: proof_reference/053b69cc9c1067f6ccb99754efe564cc63feb812ec3a0faee695e50cc1a489d3, proof_reference/25b8be66325038d957aecc65bb008fa7f1b11bd4b79a75b8a5c60207f8d020a2, source_statement/a45c8f1adf1b1dddb4d3afa53a836b0a56e663644975de6bdebb9fcd756dab1b, statement_distinct/1691e61913c18c8abf994d04399609d580ca6fba061ac3dd01e9d9497815436c, statement_distinct/797b8761697ee883b837bd16aaf81ba2499f331f1a746e026462dd5d9c45d3ac, statement_hypothesis/73d5e453b5527db8b348d170658cd6fe538600f3bbd5a0737540f7ded9fa1bb4, statement_uses/02c5ed932c47ac5aea3283cf90134f8c9ba5fbcde76d9aa66899b53776b7f299, statement_uses/5617c01d66e65e5e914dfbb996ff4458851c1eec8fdc538590084e8a4c8f64ef, statement_uses/5a59517e4ca8aa4bb0d9bf96fce1fca9b8b3861e8064af1c30bb123f296f6802, statement_uses/74d83086de874fdfcf0e6575d659d3fbfe10424620d23c500438b8a8e45f5cee, statement_uses/92a38b7836ae95937a0cf1b63c11f7c995a3ef31d3fa8bb5566b8923e159a224, statement_uses/a0089e6a6f6effaa75de41c1708d5799fd2efc17f7ca3b9b27f25765eabd5f9f, statement_uses/edf55c14158cbd112373b4be91cad2048c8b61eef2524ee5e1ab29e347e479aa, statement_uses/fe878691fdf3399cc53959c38ab591f975d6909a8aed8ba0378a1232b173a583

en · unofficial · sha256:932799f2a1c3d75003e21e3c52a096ec4e547c7da34d85ad3c688804c93896d4

con1i $p |- ( -. ps -> ph ) $= ( wn id nsyl2 ) BDZBAGECF $.

Referencing norms: proof_reference/13adb25bbb9ada610c06b5e8cea66ddb11cef5f4bb643ad9faaf0e3e271bb77b, proof_reference/18ef9abb905839b516355ae0160ec733e1345b635efbacb29408fb947151a450, proof_reference/9b98b92e9fda9bb692380cec5ddc0531c1da4c01806ad2aa94fb67464d60194d, source_statement/3d86fd7fb36e2032e122a51e41b52a6b5e2b1ca4caaddd2cd28318e634ea1d7d, statement_hypothesis/d88a59136c522d75adfd9e2cfd373c1be779ee891c9920411b98bed5818b3529, statement_uses/34ad8299c86050630a10989dbe4e79fcb3a6ffe0fb95155d72c337bafa908cbf, statement_uses/9d1af15c1cf09545bec63331092db0aabb1a83b921029a7d74398f0346e02c17, statement_uses/9eb68d7112669137fdaa29d009dc1410877956f8c5f2a6da842ed607113b1119, statement_uses/b126a2249a30beb8e99e642b8b4feba5cc5a35e091cd55318733ce1706505de5, statement_uses/c477be5b44ede6407868acaf77baa1ddb29e4fc9740e37b9dc25265bc7b84b5b, statement_uses/f357f5c8d5960df63a97bb1b80dc70ad9199162c0e705570a9aadec04c27e360

en · unofficial · sha256:7eb7bc5d05effe58aad3efec504bc44a5c5447039e7f8a0ceee87b8e9dcf8734

jarli.1 $e |- ( ( ph -> ps ) -> ch ) $.

Referencing norms: source_statement/2b81557868b94cd8494bd2d7dd5f6962e709a0c27e1b1377c2bdec41a8f9db4b

en · unofficial · sha256:b2d7af11dc6e5dfefd9b887d44ce5b8d6159a367e032d25612cbd48230a2959e

$( Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) $)

Referencing norms: statement_comment/a89a8cc41926f781207fc9d38c2a0d09794144eccf9bfa0fe909972d0dcf914e

en · unofficial · sha256:63884369f330ca920af034cd07b105704aadf66c4a60a3156f3cf97ac30af6fe

nsyl3 $p |- ( ch -> -. ph ) $= ( wn wi a1i mt2d ) CABEABFGCDHI $.

Referencing norms: proof_reference/1934405347d09be55ea0f8ff4d2ca5536814cec414f7e5b86ffd784d3d84d086, proof_reference/2c7fbf2480e9332bf0004bf2d8759531a55dc9327d322539a4575d126d0682a9, proof_reference/7b63b70f5b3808a59ab74e4b380118b339dacf4566bed61be68ab083aac0979b, proof_reference/d3c5c75def1ac040a3cd6595e1fc0065a786eba804b7d9b2d90341cc12142c5c, source_statement/9d11667fcb6ccb653e40ec95b88b5ed9506ebd0f353547cad83f63ddb8df659b, statement_hypothesis/5950e5e46bd89f01b16fa2d497ba98ec2e8ed4585eee927992953978a9834be0, statement_hypothesis/ce5fd453fe3a3c4218a021f5afd1c1128b20c2604015706c73cd3c201173e437, statement_uses/75e8fd271bb234430fb7a13014c5d31d15829e4ba3b56327484a4fe6f7bacce7, statement_uses/760fb222f98faafdedcc6e10583c9caa609868b471037fc0baf113f86666a086, statement_uses/810a842c76c934d71a86fb658ec007faef3861463a09fc92f6b1d5180d2b96a3, statement_uses/83581f3644f14feb2df2cdd5224cbc2745c9d85c81441a467091c5dd57907272, statement_uses/8c6a6e3490d8f9c37fa890556c721b4ef3f2aa006e8dfa2419d8d21df9ce2e28, statement_uses/c3273dc0cf8285a2fff0ce3bf36cf67e2627dc7ba31ad5e634fd891dcaa79aa6

en · unofficial · sha256:6ad46a2c4d7c27be2f16266e1692567a077e05f69c52d5df5d69104c98428b0c

$( Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.) $)

Referencing norms: statement_comment/fa32486eea4bd88363114ea10ce18208096fd92cf47c9541b3897374bd297d9e

en · unofficial · sha256:619f4ac9c15474a148fb9ad09d397cdbd82a7c50f984151bdb7fd26233341429

nfmodv.1 $e |- F/ y ph $.

Referencing norms: source_statement/394b766dcde125dadd9ee96fd7fc74edae5b76c3a4c187dcf180f8e3ee49f500

en · unofficial · sha256:fe764b2c71cd9fac42da463f93f1febf3b40d1de8a7775cbf4aaa6e9fae843ba

$( Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)

Referencing norms: statement_comment/83acbee4575e818a590e75a98cce8d5cc235ae9704bd1101be2e4da6bec20655

en · unofficial · sha256:3cef6f55bb3b9f229fbf8e25eb65f987d1417e2309a9351a2bfb3f3d8e95b25f

sylnbi.1 $e |- ( ph <-> ps ) $.

Referencing norms: source_statement/8ce86b920446a9948706906f93f213f58d5da517e03bb1637453550721add90f

en · unofficial · sha256:6bb889a114588bec20b9e885d6e53687bf55b39df651b9cced7ba4a7ea06c2f7

$( An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both ` ph ` and ` ps ` , since then both implications express a type of nonfreeness. See also ~ alimex . (Contributed by BJ, 12-May-2019.) $)

Referencing norms: statement_comment/8fbc99e7b1460c8ca994968e005f38b845b836078a3faa06495987e91c0849e5

en · unofficial · sha256:1c752fff879132d086a813dd0c091e8feb034481c4911e7e1c4410a078656540

$( Equality is a left-Euclidean binary relation. Uncurried (imported) form of ~ equeucl . (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.) $)

Referencing norms: statement_comment/4a616a2d83dbcc659f25812358eb17d68f22b2856c3ffe4e7c5307cbc6c9757d

en · unofficial · sha256:213f7d688c55814a85054ca7b29892dfce6567863a8d3c79ef24885143d7e6af

bicom $p |- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) $= ( wb bicom1 impbii ) ABCBACABDBADE $.

Referencing norms: proof_reference/453ff43b019f73abecb60f8c1b6bfb9520e3c05f430d65e9577cb5026710d912, proof_reference/6bd3db84002946715efa863623074c01fd422b57157c6afa9feed6ff9406b2ef, proof_reference/a8eaeb3f1c8f9e3b5e817a99634ec809f19bea6ca8cf83a6aeb877bff3aa21b3, source_statement/b337ea999cbb846f724bab50d7cfdfda3aaac57d13f4c94b6223fddaa09dfb0c, statement_uses/285b80365e6badf885f7267baf99f487cec27b3369f6481af2a03c76c5c321f3, statement_uses/2a7e2fc7cb79f2227c5a4e8372dd696301cd222b4b631a248eff97f89ec09d21, statement_uses/482bc0eca180e65c2bdb90a31e3c24bca4e044f877c9ad2339b26174c0c47d95, statement_uses/79a268a973d7c36e31a5b216f15d142f61eff9e5dffc44b14457493dc1a06eff, statement_uses/f16e96501cd93281854a74d3f15dbc8c725fb6c946909a550144b6da06858ffb

en · unofficial · sha256:b1baa90fbf28fd896ee0ef723857da2824cd7a7aa56910829ee8dbf5bc9b4237

pm2.01 $p |- ( ( ph -> -. ph ) -> -. ph ) $= ( wn id ja ) AABZEECZFD $.

Referencing norms: proof_reference/24b2a48b03cf19c9a10bf3d4992f485716171f10f6ec21222c127034334c63a3, proof_reference/65ebab33d29650451616915aa15a7438821df633ed9027bf5c7b1dfff3aa3ad1, proof_reference/fdf02a1ea9cf11f36aa8e6dbc690acbe451d024a21dcae4783c93dd2ffa26af7, source_statement/d320d0d733a87085f8cb10762e850f20dd7a97e99c5467c57e0dd355d9bd8cf2, statement_uses/084419c4bfad41bfb8d3687ce9a18239c7addd506c2c168043285ef88bf0740f, statement_uses/3035d62e55c9d467a58b81cf2b16da9c33da4b3c9680e802a5cc0d7fa45c1647, statement_uses/4274b3e1c383ee54b949e179952a423fe3ca2f6fa8e83912ed2f489cf0f7eec7, statement_uses/6d5ed8a8af2642e53b8997996375615683c4230bdb9733b07725f0dca1a18818, statement_uses/aba0db899f6813eae8abd6debe678b978c2b3090fbc776602a552720a3c20e8f

en · unofficial · sha256:24586391377bbd9b1fe5c671d8f9c6d1403fc27dda9a460945bab7f00cdb7588

df-mo $a |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $.

Referencing norms: source_statement/eab2bd3f1a77959d8ea2cee1356d18278cff7d406f8ff3eeb29c8168213b28b2, statement_distinct/042e6936f7c9a6e9c724154e1064dbb6f117afd1fb7138515fd4486df7746541, statement_distinct/150dee0d281ef2589516d9f2929a900b24fc6e10c9f7a8de11ff56873ba67bd2, statement_distinct/387718fd710e5f7adf85c553032c2bbf6d5b21234b877c25f195d8e374c29c02, statement_distinct/819e1a8a2835865f2af39c868621c0cc926560649a3614b0b7012ec267b8b20e, statement_distinct/d005a4bbfbf2bdb9d5878a4c174f1fb7595d288b2c0e276aa14c66f5b2901336, statement_hypothesis/97bc3c4d3eaff23db259478629b01e172cfaf84ac15e5e2ff154682a912c1f32, statement_uses/01cff5dca79e8787d88e90f0d2a362a882bdd57c55ba340df314f322406a4274, statement_uses/036fab8dad04e5342eef2957829d48367f4600ea8cf5d1e95cf99298e3162a18, statement_uses/08f6f317fe8629f1631cd8ec68d27ded0efe7af95060aefae3c61d244b5c43f5, statement_uses/0b130b29b1fee8b598025866c04b13a531ba9c7224095854368bd96f1ed90d2f, statement_uses/0b45b450dd5157c3c73f480b6606ee3f6e04079774e7280f82748192bdabb8d9, statement_uses/2b6bcc3e25bc297e6ac427ac64f7bd6f7897a4e19b40dd73fe63e070877b0124, statement_uses/2c7a12b9be1937c8d7e3f6cf34b46b28a6557ab200ca929170d7bf95e54b5072, statement_uses/44d26c39b1dad4baa014b2b90a9119081c69660ee8dca4b532cb5bb4ce5463c3, statement_uses/7acea52120196d3e4147f2c6a8a8d1c6ee3212eb0c83bacaf3d2a2cf6121442c, statement_uses/c748656c8e10ee0c9e8effb532be545cb97de7c836575650e1f3518b23c127a8, statement_uses/de0e963ee0bdd555d63e49f49b242c9ff04fc60711ed2a699a83d458dfe33aca

en · unofficial · sha256:ec195d675bb42f53f7bacb08ddbdb338d37e56070049fedd48844a2f5c65fffe

$( Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also ~ biid . An early mention of this law can be found in Aristotle, _Metaphysics_, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.) $)

Referencing norms: statement_comment/602d7d220f4326dfadaf28f0501c09c7e880a9a4e7d91c22a2bfa8da46cb147a

en · unofficial · sha256:a99f8fa938e31ce18ce81ea5d4be6be899e4285ce365b4d5c6d712880e72cdf1

ax-4 $a |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.

Referencing norms: source_statement/a4483897d1d2b63e1cbd42607ccadd2e5e9d450614657dbb4eff1d78490e4dc1, statement_axiom/5a2f6ee86b542a1707d7e1d1de57f9948fe60dac2f2cead027f2a224cc80c5ee, statement_uses/2b37f2fbe41d92144807309e4a80af9ec24e8615b4a058839ceb8c0ba065796e, statement_uses/5b351419fe91f6c6fe7c787f609f41c3aa8f54a03152ad61966632007d51e385, statement_uses/8be96152c1d974ededb4f75a336486e5af76bcb9eb2838324648e7360ab7eb01, statement_uses/8c02aab66796e82fdcedeaf2a2cb9cfd58eb5771f45610408844d6def20cba29, statement_uses/b1881bbaffe2fa6c877cbcc34e4ed4241e9f0ebfdac2fca6261a09bc1773081d, statement_uses/d08f0ec64953384b5665ff6e45969409805930a76c3dfbfbf46dc176967e5f16, statement_uses/d9aeadb91c2b8e5864bbc82411217cc8f552061730f12a664b0e7bb7971447f8

en · unofficial · sha256:e2886be0487772974e67519ea58bc29a0c9a9a87c96f90a295eb29d49868cb96

mof $p |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $= ( vz wmo weq wi wal wex dfmo nfv nfim equequ2 imbi2d albidv cbvexv1 bitri nfal ) ABFABEGZHZBIZEJABCGZHZBIZCJABEKUBUEECUACBATCDTCLMSUEELECGZUAUDBUFT UCAECBNOPQR $.

Referencing norms: proof_reference/06777edba598ba231fb65bc3ef24da8bb309b477bc13fbbb0c89d21d1bec66c9, proof_reference/16683e552f703bf97cca8c1bca3581adfd5aeff60e13ad7fd64671c28629e00f, proof_reference/23cea2409b946cc8aae78bf22716c2869af51780495252c99e489e754e2d7007, proof_reference/36272eb7f8218c951e5b83f7004159ea8fc40b1fe53fb2526184b167735cd5db, proof_reference/38b2a20cfc7db7a5fec07fdb7519cd65f94ae2b56fc6ac25a0c163b64f636546, proof_reference/520ae204743657e6e6d5db04d1d249233a1a320689643ead1971e80f1b66f304, proof_reference/579521c370ebf9f4951e2afc1755cddb26ed2922e348ab52bf61d272bd6eb110, proof_reference/6653829a9c0669d114587c43419a8181859b8d5978b1ad738c2729ca706e8bbe, proof_reference/6b6b08c34d74db18fa3d5efd5e62a54eb1c259c80fa672be65a2841d3fbb552b, proof_reference/792da1e95d3ad91bfb4eda248048ad7c6d5452060c36a913b63d3fdee6289ab4, proof_reference/937603e94e443b062892a41cb8b942a03f2d2326d28a8de0e2a3a480e65f5603, proof_reference/976401cd868f810e2bd9542ec272fb9f78eadc80f696d75429cb96c6b16b44cb, proof_reference/9ec55e3207f81982aa66c2ef1a0bc6323880f697501eb061cba9f19108e2c86b, proof_reference/df10f68ad9b86129412a5aea77bac1138159db29856c25724de023509b4133e0, proof_reference/f5436b8b5af879eda76033e2d2299edcf6834fdc37da186458d85d9809c73953, source_statement/12f9552292d804d4be4bfcb24a546e05c14cba0abebb0dcd73311e69bd605d19, statement_distinct/ff0dd9cbf3d04e690af528024b499576960bbf9a3d58e0a2de6939e11edb8afe, statement_hypothesis/051f0e38cc5b9f040475c59c552bd7a9e70e609d68c67dc9b1476151c7d28798, statement_uses/3435cbcc76d212656b5b78ba3369997e06de766d9989371eb851582d8f915f59, statement_uses/4f7bcabce2f71e9cf1b15f5f25dc3c4f998552fcf473d433a6246f35c4d2f7f6, statement_uses/504cbd4ab496d3221b7db7e408f9f672596f2b1daada884d2b909f9cc39ec8f0, statement_uses/5907275df134b7622a34255b6a96fad69bd872d59398c06c946ad2ee3954da42, statement_uses/67d0a31b50d64b8ccac1762dbc6d3e034d1ec90b76438d5fee460bb6555004b8, statement_uses/708c57d908bf3e91a838fc28c6938cbfdae3cb78579210a6722b5e2a51f44f86, statement_uses/7511eb056a35572835eb26288ebb23da7c78a1ee8d759336da85c203061f6032, statement_uses/96ff15d320bb3d90c987ab7676bfa53dac541b4adc2014f967b19858881cb63d, statement_uses/c52f95441b8097d4951afd0608dbb8374bf6608a2c142973b2c2f85e71430574, statement_uses/c7bb2c31b0fa6aa64d3954ebb91ada787aad3ceaede0b8cda283f0dacfa04555, statement_uses/ed9240b52669bedf2a1c271da19cca7f61b7d3f5d6e973e075d91bca34601e29

en · unofficial · sha256:e673e65accbcca5329c2e98e77da30c059de0079c64b3afb39651243cb3a11f5

nfn.1 $e |- F/ x ph $.

Referencing norms: source_statement/1df495dc4a21db3cce5a5dc81340891c6a672f3bcb67a7cc194bfd14e8fe809b

en · unofficial · sha256:f45be8919a926af89778062a4ce363b2977a1fd3a760596f4d59f13089af9c57

pm2.65d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.

Referencing norms: source_statement/feee31a2aa316f79ee8c0c73eaa65ee28fb75dd46d5ae141fe35c712b4633fcd

en · unofficial · sha256:0b01520eb61cfee345e2da42de521c11d0b389b2be9b78428679fc98ad298fbd

ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $= ( expcomd impd ) ACBDABCDEFG $.

Referencing norms: proof_reference/35455b1f5244c492b918306840e634e28b67e524af6af37e7881fdad29114500, proof_reference/5cc47a33d071145913b7900b56c4b54b605df15ae75e969829562cef09b0563c, source_statement/70c40ebed376d5e68e13fef21c86a5dc220fdacbddb0ecf66aa982f04795c074, statement_hypothesis/535707d6839935e630161975ab4cca13da11b48f07ba07668c5d948c2b20d1ff, statement_uses/1c4eab0ad88c0345f3a8c8d0fc49293471ecf5fea8594ca6a83762b8378a178c, statement_uses/2ade8cce685b07d5ab6ab4324700c4cc97f26d2251f6f4c2d4efb04e6c4db9df, statement_uses/384ebf03a9ec398cc27a74a47097924c216329e45c1c1fce20df3f90f9b0a73d, statement_uses/3909d777d6590328acd1ee20fe8cecc9aa11d80c5b225305d30cc35626768e44, statement_uses/5e2b7baf9608abf5c1ac97e026d5ba1a7f43ef7d0f2fbcacc6cfff70821e4d7c, statement_uses/67521c8fcfb1328beac26fbc4bb1107f1563db78714ab223a3638d9da6d84994, statement_uses/9e72f54a51752d772eb07269a4631005a796d2efd23bc5f6b0759f89a10b5c5a, statement_uses/f3c7bd53ebc4677d9af9b705ba22fbe79f6e596be202ddd62ccd0921f12d644b

en · unofficial · sha256:6b435675383c4576a8111fe4659e1bcb298d2a3f6606aa1f506ef33962c66055

$( A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.) $)

Referencing norms: statement_comment/ed56cdc48d7477b4db87462706ec6a1a5e5a4628d14f63ede94a2d382e05ee3a

en · unofficial · sha256:91d40bf115f50390a32b7bb4969c7fd37bcb30ed8684377aff65f79b2942563e

albii.1 $e |- ( ph <-> ps ) $.

Referencing norms: source_statement/3487095c7833498c6b9ae30159cdec07fb81f57bf0eb01fa6d26b08c490eb64a

en · unofficial · sha256:c21684b98d16b46cc8dff962add1352b61de3d6e90ce4d14eea6fae2c89772bd

2a1i $p |- ( ps -> ( ch -> ph ) ) $= ( wi a1i ) CAEBACDFF $.

Referencing norms: proof_reference/107ad272d04a078a9882e5e2db411c7d78d47d34c33d8b0318f167deaf2c8fd2, proof_reference/50104b86652c4c06cf2b6c5cd60dfd59b155bb3cd8987b2e11d411fe13e4d3b6, source_statement/a178697b08c77cce0728f44687984d691512bd3ea8ed7ab6ffe725b4d4cb3dfd, statement_hypothesis/d8ae05594ebbed0b40dea07479d87b86b8b3e40fa3cd2325827de5e16fc64cb7, statement_uses/0da1047ee4878590bdca62c6aeb912cf7c272d000f371a1581eb0a6ccb83327b, statement_uses/14cc6eb4483cd8839b0ce7106a72079479547170e7a89971aebe121cfa5161cd, statement_uses/d94f9db0bdd6aecfce637f8fb76a0fd87c4be931bf6571f2fee2f25bb626c6fe, statement_uses/ec845e2b7cc6c1132328a5e0ebe79963c67a9696ac3d7285f53d03049c7d4d99, statement_uses/f99b4be452b6f941c04c6e41e2d4901fb37022e5b44635a6f1eab46b030a3af9, statement_uses/fc642be94b162e624ccff2a008f8606abbc6cffc9e4fa22438c5924a1f34f6ec

en · unofficial · sha256:a9eca38e15b22f5742d000a2ae14ae22f7232d15fed4d64a29217cfc94bdd009

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Axiom scheme ax-10 (Quantified Negation) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/becbc11fa9ca450ecda78497578a311ec1503f313ce4d1a5c904968aab50444f, section_title/84d89509cf5c571ca986626c58edfb7b399205681ab85b8d487924116521809f

en · unofficial · sha256:a476df3271cf9ec51b7ba455b825454987b0bac087a9889f05fa6473888fc13a

3bitrd.3 $e |- ( ph -> ( th <-> ta ) ) $.

Referencing norms: source_statement/e17b0c55db47e123522ec6dc92c55bba787e8390ae57896720019466c50097bd

en · unofficial · sha256:7f2f7c9bcc950e89b02bb4aa75f2e573de448681e6df8ee9a39829f09fe2cfc2

$( Conversion of implicit substitution to explicit substitution. Version of ~ sbie and ~ sbiev with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Aug-2025.) $)

Referencing norms: statement_comment/dd954b0b81a0f125bfda8fa0ed7101231e1fc1cc0e8234622a3d590f80b27d6d

en · unofficial · sha256:ebff9d66c57d5f5754c7ca9a2ae158f5f2e28731c9ecd20df45558bdea0db886

$( Version of ~ sp when ` x ` does not occur in ` ph ` . Converse of ~ ax-5 . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten ~ 19.3v . (Revised by Wolf Lammen, 20-Oct-2023.) $)

Referencing norms: statement_comment/fe1b9241bcf8436780aaa378de0abc3292ba0d6e3e92583ad9f4225674760d55

en · unofficial · sha256:ef20d1fe078d6338d10e2a868a168cf5e69f49e723b279e0d7aad6a231abe03f

$( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Introduce the Axiom of Union =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $)

Referencing norms: section_parent/344d4f4825525d1a43035eb203238ab543a61c549e50506f98d6879d40cba477, section_title/1bf90c623589ea1b781eeff048fb2cb05685e781ccaa14ce561d093a286b855f

en · unofficial · sha256:86140e6b2626e960cb8bbf6507e5ac441d1ebc02faa9229126581856b8ea340e

pm2.21dd.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/5e8d620246dbb13dda10b93714053740a98d59a5add21b94220131c9479d63c0

en · unofficial · sha256:40ae27409b187b7ef7172212d73eb17d5ef143941f977093e2522c62eb699926

anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.

Referencing norms: source_statement/d4d10d3bbdb8f99e0cde8591bb71d7cede97f8cb07d7f8d1608cf757ee786ada

en · unofficial · sha256:795f6e3f36bcce03d1c67ac1d0e05aaeba7f7bff50444193ecc841d2ccbb7071

$( A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) $)

Referencing norms: statement_comment/78ba1d6b5fc68716fbb13e32640f1642929a13246d6e8391e0815a234b1021ad

en · unofficial · sha256:1c4ad0364cf97d48476a6c77d21eeb9260637ffb4d58bc647b624ad4197c65e5

conax1 $p |- ( -. ( ph -> ps ) -> -. ps ) $= ( wi ax-1 con3i ) BABCBADE $.

Referencing norms: proof_reference/0f709cd42dbf0df0a69637a8f682dbea35d613b87796a6e8de80930ab583d397, proof_reference/5941a49d8f874f3408a5d0a4f18a3e8b60fe4e5701f1f93cc978bd672f3906bf, proof_reference/916c88e2fe6c44cd6d862da207e89e64e42688f4ef4b40ab8a36adf8384ee826, source_statement/56fb9913fea35f566d6282bdd40bf1d81b5dbeb709238c998ba5a956419d0787, statement_uses/0cb5122edb267333dc28bc40e016368531158e8005851ecbf6c34e0c6b8c1e65, statement_uses/34ac1ddab07aa2949bf03a471ca38945f9f992d4c839d8aabcc0e71777fc2df5, statement_uses/5931a68f6493545c0027581b8bd53f5944bb11b98efbfdc9d43fcbf10365de7b, statement_uses/6e70b5013ad3036c52ba5cc155af5d9c4dc672ebfe0ae6b4177177396011f152, statement_uses/7097c36f8823be6298a7ca518a53f988bb2613ef552d828fcfc0171e483c2a09, statement_uses/8b191be488e30807637d5974980d0b336958258b4a0f6d218f06322b2493dae1

en · unofficial · sha256:d262feb773c27baeda5b90a22961db86c8df65895ffa4f181c0789bfcb260934

bitr4d.2 $e |- ( ph -> ( th <-> ch ) ) $.

Referencing norms: source_statement/1d807d54ecaba6acb6960dd92da11b00a903ec8fc397969b1af274426bafc44b

en · unofficial · sha256:9ebf940fae4f30401b6337bf8146d95d107c4b4c3750c0008e8a98aef29e65d4

$( Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.) $)

Referencing norms: statement_comment/fd408126bc5a9d6b30a987c700fafcba090138b15d6900fa5d9fcb80b8c42aa4

en · unofficial · sha256:57b7451d6e3eded604d1d9a32632eb728c2d2b8d0acf0da21388bfb9cb2a0a7e

exlimiiv $p |- ps $= ( wex exlimiv ax-mp ) ACFBEABCDGH $.

Referencing norms: proof_reference/2f716d229e46298e29f128e68584b9668de9d9c0f54fbf563c94e1ed46027a53, proof_reference/975a18a51bdf6aa567751fff3a699ac520195ec3e1afad95a740315f920acdf2, proof_reference/da1eb0d430a1334c219472c01ddfd4f85e75af75e6350a734a1c37739b17a83a, source_statement/8361771dec9342deb8d9446113a6da43f8934d4855aeb9db696246320a30ad2d, statement_distinct/47d11e84305c3ec1119f9b8519fcb9472aa41dd78b530ae070f0d0ca54a4a1e6, statement_hypothesis/a61601fdd5a2efd529e26ebaae1b696d56af301b2c875b95c639441e45da78bd, statement_hypothesis/d07515f10140f1331423fe8e7355a93b6447b88fd24a0378dc1e2b94f77af88b, statement_uses/2731f3c6fbdf0b4fa2a81a28bb4c2e81143b02daf2b532d5d8c3b0a39da064af

en · unofficial · sha256:67479666d61ac4d335b540be5adea140d11d7e6397ab93c703b4185e56ca5e64

mp2.2 $e |- ps $.

Referencing norms: source_statement/62f09aecaf58bc24c2f2eb5062f1d4ce23b447b2fcde5d4e887876ac4ec19127

en · unofficial · sha256:5cfab423d067d16adcc5cbdae3a6e195c77357b186a574d6ea6f63b105d428a8

vx.cv $f setvar x $.

Referencing norms: source_statement/4dfba83d6afc005e2b67afc0c22625ced5e17682faa14563f09fa0183baafa1b

en · unofficial · sha256:2cb700e1d79478615cb53267d99bc96235f0ecd8ca0d6667c984e1531087d8ec

bianass $p |- ( ( et /\ ph ) <-> ( ( et /\ ps ) /\ ch ) ) $= ( wa anbi2i anass bitr4i ) DAFDBCFZFDBFCFAJDEGDBCHI $.

Referencing norms: proof_reference/28978238abe0c5ce63751816a7f07b080fa0cc10871cda984175d937c685f6af, proof_reference/38809672d8381291b1eada03f0b68b8732564bdb557344f8cfa1ec6180e5a51e, proof_reference/408abb7f1b802888c9f70ba6c52474e80aefac956f2a4a430d89fcbba9229921, proof_reference/711a38e2b7dcf7b79d0de238fff9ddfcdc15b64f4e43381f19f94967a1fb5a5d, source_statement/424cc9d1bc2012a376e7f880758237b0be715e370bd7d364240af9bd26423807, statement_hypothesis/bde4d80203a04e686bb565989d73ec0e2e59d03667bd5e996dc760bb70c13e26, statement_uses/1189598a483d5b736b38b6203dbc8ca5f454bcabdd0bbf428c99f48a03b19d08, statement_uses/1dc064a8335b76266c068bcf1329422a85acbdfb0512f598a3fad60846499a85, statement_uses/26e6688a6be8751f0117daefa8bbd059feb5189043f135f9ebd66d6df03b3f4d, statement_uses/3b758abb00c9da1bb68927a2bfa8433ec5e8938dce0c5d2c5a81a19ece66f94c, statement_uses/b66588b7f365f79af6332837ac2beb4ea763f344235bf41535403d66ead217c7, statement_uses/d751923fc5610c7358b749c9f4b86fb95cafb96df0f07a4a50fab6044afbcdf1, statement_uses/e49efaef258a2c6d2a7d95cf385d7be9ed3913f75ec96fe2b5617384756863f9, statement_uses/e8432da48a8cbf2ade07ffb2f9679e968208bf22f3242306f1506473b233d79b

en · unofficial · sha256:0b64fe0878a1c6e576ed37d1c230c0058f193e15c6763b5ada60edc726e0ff9f

3bitr2i $p |- ( ph <-> th ) $= ( bitr4i bitri ) ACDABCEFHGI $.

Referencing norms: proof_reference/14bbbebf9ff690b8f82275c4edf6c11c8e27a668efb5dc83ba3809d0a4e68ebf, proof_reference/f2ae438189f96f5f19f9f243e8cc68b1f0cc2ab9d2b84ed97f94ddc44e98698b, source_statement/9761eb8b024235ba6a5dd1952c6177e97fe69c2ac08f8c0ba315c6df0e0f522d, statement_hypothesis/38a21f9dceb46231e3988fef081899531f5f48394afee83ec057682f64bd73a8, statement_hypothesis/7e9aa8ff4774ed4dd22fefe5081f11647a34a076e79833a3d7d973f016591911, statement_hypothesis/ecd5cf7116054b47da090d679b5395bfd80154cb237d74ac325d3be8306d2335, statement_uses/221816279335f519fc24627abeecde4642b447872ef32a1a19759cbc3f0bcd9d, statement_uses/2a92a9fb95168b6c939b29437dc8589fd996455fd62741c69e3c3b16c0a7bcdf, statement_uses/3ec01be2aca44ae36bc83367230099c08f08007acf80135926a193391a12adcb, statement_uses/67c573665d27ea8bd4a96f127741744cd10979e527cfc8500b8b80995bef2d5c, statement_uses/956b6010424981ce082cb5c4be5c9f9119469810fc891f2b96566e85058b5057

en · unofficial · sha256:61e2c8025daafa08c90a6402a7edf99b2cd80d34dba589348ce2eae6f9ed48c4

mt4i.2 $e |- ( ph -> ( -. ps -> -. ch ) ) $.

Referencing norms: source_statement/a14fc3200be96cee4bad2eedf98f66f53dc6bfc2245a9c2354830838553c830a

en · unofficial · sha256:38b9ac1b332c60f8a49bf597354e81621a4d7fbb94a907a159439967303957f8

$( Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.) $)

Referencing norms: statement_comment/fb38a353aea83a398af00fa4b10da72c44e7b97e5213a6b0db0e78a96f7006fe

en · unofficial · sha256:7eda6197999496ef91b8e16ef7b4db3434ca1b8402f4bc334bd7ec5a14025339

spsbim $p |- ( A. x ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) ) $= ( wi wal wsb stdpc4 sbi1 syl ) ABEZCFKCDGACDGBCDGEKCDHABCDIJ $.

Referencing norms: proof_reference/055fcef25baf1bd77ddcf65fba5fdc3ada98dff62d00588fe6d0b0d50b005444, proof_reference/45d5875830286edc2122ea9751cd874337a578852c1e261736c055068d026efa, proof_reference/537b95eafcc684e343801ea8f5c02536776483a67db9fe86a41235f863a60cd4, proof_reference/5b5f5bf250fbf5ef20f10411b929f416d3ba4611e60b3f8fe4fc78d11ebaaf8c, proof_reference/63de7ef2df596ddd29eb23626facda8dd08734687a8116b2d3df527f06a3d022, proof_reference/d0d6f13a0efcefe1950dc5d4bc1c2468c1e3880cb8ffb6d9b321b4e241b814d3, source_statement/5b9e3d8d13cfe604b15d3e7f9524373d5b910ae6d21913d4b8e60135b3bbedca, statement_uses/42bd7e14df964def30037a61e18850bfaa7a3d79d759ed169175f78ba0f6962e, statement_uses/4b5f7b9c7a94c9c8ba36fe723fb81b62ee09b8c154820ddeab8eefe1ac9e8473, statement_uses/580699211f9d753e8eb2c307f2ede077fe29ad8efadc8205416dafe353e95fba, statement_uses/64e4568b8f494d153eae8835b4b19722a47d76c58d5b23da9455dbcd9842c38e, statement_uses/6cd13e8753e0aa1a07237d04a20c327acbd9bbc8e740b30074800b0947de5fbe, statement_uses/74907fbca07c10f97ebd930f1b23ef67f71f54d8b72e35b73196c9d47de06b3e, statement_uses/8c91c3eeca961caad24b22f00fc471bf9db2a5c0798a929765e36a2b84d76c8a, statement_uses/ac9b03607b9e3b05c458dc790cdd1762b6c12997af17337108642b9c7c86038d, statement_uses/cca64b8944633f363d3a13c659118247e055dc7714f8e1df9ccc9456b87e7a27, statement_uses/f3a0261d952d1fd10c7527d47b2518d2b9dfc35a4923dd6308f0efe32182b5f5, statement_uses/ff48df88c425d01dc1b7d8b110c6eea62f07645be97d0b8906a0d2056d35c87a

en · unofficial · sha256:e4322d5218f7d5092e14c7f7727a9235e1f298049bf64bb3be7fc8f8049a6c7b

biimtrrdi $p |- ( ph -> ( ps -> th ) ) $= ( biimprd syl6 ) ABCDACBEGFH $.

Referencing norms: proof_reference/422aee5251d28b2178b5ecac5bedbf274d7e0e5da9d186acbb5e20c3b7df928f, proof_reference/7152511175a4577133b2d9b6ede68f4503cf6d492ec764bf0ba50c2f682039bd, source_statement/8029b22dc50cf6e0a15228554b4b918167aff967af5745e604af4112f0fb2129, statement_hypothesis/0c110d35b4b856f8d45a167d33b6598baa23d1bb2ba54be6e8bc03ac83ae6da1, statement_hypothesis/165fe24acb65b3e9e2818515a295ec467db83eb40e382183d9bc00a0c72d640c, statement_uses/37dad57c574e997e64a43b535840faa6db7f3af4d2249c95a8b8deb8c1260a52, statement_uses/38b3bdaafe4d34ec23b40b206e1cef366b7454ce7b01e5d46b1a785c946d8925, statement_uses/4f273edd89e88170789cd5a0672a31ea5785fb71e11e947b9c62080019327b71, statement_uses/835dec3aa2b6d048373722f4edbf5b8fac7502a98dba79c83025ef9c4678bcf9, statement_uses/8e01bf7351fba7731ee88986a584d3353f6ce500690a8e05ed8a497ce67a305b, statement_uses/94f28c7536bf222c8bb9cc975db53e0c179bbdd90657f063a6ca5871a4ec8b02

en · unofficial · sha256:b4d31592dc557ea71c7c7a9c42ca0058175148b0a94047bcc29dd66b87b2c3e7

$( Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/91fe82265bfdac301591ea8d53d7e937b4677cb3aac3ef0920ee3de41844c016

en · unofficial · sha256:2d0a243fe81f5f5c8c7b19e8391858d3ef97e8b5bca2c3ddd344ad304e8ec12e

$( Contrapositive of ~ ax-1 . (Contributed by BJ, 28-Oct-2023.) $)

Referencing norms: statement_comment/273c62222960f58b2556a1984c463c26b02c89542a6a14e9089c3e72afb52e26

en · unofficial · sha256:ffa64382697ff558e8629152703d896a4587f0249660f98650fdfe76db91fa77

alrimdh $p |- ( ph -> ( ps -> A. x ch ) ) $= ( wal alimdh syl5 ) BBDHACDHFABCDEGIJ $.

Referencing norms: proof_reference/3145228574893ebb407a0052a7741d9213e24e2ff008587e932032499c2e8d7b, proof_reference/5df015138a48dbcaf16f819f3fdf17d5923a8ce52fd702371d862e4df52ee12e, proof_reference/aecf814fdd22630a00ff222e6f75bbdfd1dfc71970bae6d1cb248727ed95f0f1, source_statement/a5d34a1a6f493769e9011aed6af2246dfc963b3e3d49bb15c22e82fb3b125472, statement_hypothesis/70f951725369e7b41c1a0acf209a09fe202814b2694b1e54dd8a5bd7aafc8275, statement_hypothesis/c523a1795fc05bd32e9e70097e701180278182dd5852a983d8566460ccdd34bd, statement_hypothesis/d30972f8f0da30154fb6b673e10b5f86642fe59efa6f27ed4c45e26ff27899b5, statement_uses/27ba8d1bec74862c3b8be8156374f8a0a4a8108ef83800fab8b67e2e17df9438, statement_uses/2d26eb1daed18ff9037ed75fcb0e1188535e1da1b9b0e40faefea52a35072f39, statement_uses/2eb91d33742d9849a33e7e3d7c25172c83b8d0294316958c82cf5a324ef5c5d8, statement_uses/5448b362a5f837a78e7378d491343394b9feb052b06799f5d9e84c286da3bd25, statement_uses/b9b0788dfe1dedf20619f785edadb288693a309510335544e88951dbe41e2776, statement_uses/bac4a954e68fef81cc8421a94d3494b37e7d5f341919ffe3351fe42b43af1d9f, statement_uses/c98c2ec66d4008baa5ab2d2bcc9eb4ffd61085116e1f303ecd6dfee58f3ec7cc, statement_uses/f1773476d81173cfc674798ab2aa1d83a5a866af1799db6d4fc148ff334889da

en · unofficial · sha256:c4c7280da93458c8dcb082593e63400ce96d68643f4a89834ac5988afd49482c

$( Alias for ~ ax-3 to be used instead of it for labeling consistency. Its associated inference is ~ con4i and its associated deduction is ~ con4d . (Contributed by BJ, 24-Dec-2020.) $)

Referencing norms: statement_comment/6788edeba435360594eb06a1720ee12091b111f05b8f7800fc416435c8e609fb

en · unofficial · sha256:c787a3ee643b8c7a1a6905acb472c9a6a8c50a3b0c27cedb1dc7130b4c8e797b

$( Proof of ~ ax-8 from ~ ax8v1 and ~ ax8v2 , proving sufficiency of the conjunction of the latter two weakened versions of ~ ax8v , which is itself a weakened version of ~ ax-8 . (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) $)

Referencing norms: statement_comment/c4f8eed2755938ad51bda9c60011e3abe1547ecb0d0b12ecd2ab266fe89b6ef4

en · unofficial · sha256:2c0741bc401ecb2c04a8851ec756addfa7a0ce649d8e18008f9ee2c6f7c40255

bitr3di.1 $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/ab7b8eaec5510a3b077f15c9b4bbfa31b0b4b83662fa7b2be1997f55c90eaf42

en · unofficial · sha256:4858c3135da7c199c06a1e91b52b314e24c6d93a34e3252a68b8c35d3562813e

$( A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.) $)

Referencing norms: statement_comment/76e93ce8faa6b8d0eb3db3979068c196851204d30f957954d2f49475912b43ae

en · unofficial · sha256:d1e84e15685a79c8e6afcff0fafe69e2b1fad80287bd75a06d6b2c2d8e5db014

mp1i.1 $e |- ph $.

Referencing norms: source_statement/829355a8abd1d8901e3efd440f895f8b816e14f9b3b731f0c2cb283d05dfeafc

en · unofficial · sha256:0d79661a0e58625926c2497c8e08d97ab40337b77033f50b0df58019f121dcc4

eleqtrri.1 $e |- A e. B $.

Referencing norms: source_statement/19ded55f74e0072375f2e0d66ee92a2b1e7f08b618278c44b1bcbfaba258d9a2

en · unofficial · sha256:375cb893953709571a9064dbec598d097335b1f6d54c3279aa52fd3e61913e5e

alimd.1 $e |- F/ x ph $.

Referencing norms: source_statement/6054025026100f92c1bfa6b5cba6f16f0a5a1ec8aaa8d82007edf43e6a5e1fdd

en · unofficial · sha256:b4162b1292ed2378ee6f8a05d458c3148ba13d1907af13dc8c7919999dea9c27

$( Deduction associated with ~ nfnt . (Contributed by Mario Carneiro, 24-Sep-2016.) $)

Referencing norms: statement_comment/37ba5977a71c1cae678853105af443eccd2948d14de8f1a99d91781dbf0e5e56

en · unofficial · sha256:ea7424d1071691ad72b4f220cac45f9c9a1b73e4352a8180fa063125fdfd6f0c

$( Deduction form of ~ bitri . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) $)

Referencing norms: statement_comment/b3847f779cf8f467eda08e6aceca79d829f7790c8d31ab01016fb278b43104f8

en · unofficial · sha256:d06e18c45d7a46d30d9e1a7144e8ed786217969dc2ca5e56eff3b73dd930731e

imbi2i $p |- ( ( ch -> ph ) <-> ( ch -> ps ) ) $= ( wb a1i pm5.74i ) CABABECDFG $.

Referencing norms: proof_reference/99de1c09860215ad09b3e702a6ddb3b2f0e77a02685b47dbd62191eaee8d5cc7, proof_reference/b3f9a126917de67997bd137f6a25fde5b8c9e3a92257d289b2a1d5a63d79d3a2, proof_reference/f835cc13e1b67dd3a765facd95e5b212b559afe048af97602386b5a3a4a08858, source_statement/58e1ee887b61b2f8acb402fe2fcf0d5e9885630f91611621fc7a1393e7ae7672, statement_hypothesis/5aab84cd453dde66fe6a4bd19191ed017d79dd752198c1eddc44c5ee5c440f96, statement_uses/6722b2678ac73edb47f330f8371daf6fe746b6f238672f8ef8516e27e6da1b70, statement_uses/6b3e6bc688eb81353e4713a6ee23de9cd915fc55d5dcb135f1e998739176cf3b, statement_uses/7cb717a38d0c39e0c6019a285b3d409aa4fe5834f17925794c3029dd50808773, statement_uses/b0af8fd8e55ae9875bdfa59018a8cb22a272c50c4f82808d25f834cce02930f5, statement_uses/b75bea50701b8f4c51e0f1af41038a1c1fb1babc5d1db25b2fa4722422e1ff06, statement_uses/c862ae95dff2a65324a70f3c1843cdd17d9b280941a581f0dbf15f9f95ecea06, statement_uses/e0d7f1fdafb380b44763a6cea4720778cffcc4b0536f53cd9ef4ce6975e1d9cb

en · unofficial · sha256:2511a4f9603492e54659785bad240c229c5c794d8c87bca4f0cf013765dcb94e

$( Conjoin antecedents and consequents of two premises. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.) $)

Referencing norms: statement_comment/bdb913f75129ad071e74d71366c0795968c468a12a41784adf2fc4c2cb40422f

en · unofficial · sha256:b471507c72ae0d12e962f46b770d46fbc0309ab2a709093c6df6d6b9c48a34f2

$( Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)

Referencing norms: statement_comment/8f9d3c95f42a41489971b5e17506fb234adca88be9fd12536c7ac4541f9eb131

en · unofficial · sha256:fe236d9c26041aed0cc463fafb58473944f8ba99e55fce89064375911f5cc858

con1bid.1 $e |- ( ph -> ( -. ps <-> ch ) ) $.

Referencing norms: source_statement/df6f3f1ba9d53f908ddc8d3206bb67a386987130c33fb7b596d25fee6bffa31f

en · unofficial · sha256:41984df572ce9002239855fc5c0a02ca53bf73e60aa8555f57a6617e4359e76e

$( Exportation theorem ~ pm3.3 (closed form of ~ ex ) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Garrett Katz, 25-May-2026.) $)

Referencing norms: statement_comment/77260680c7c6a11870a65869c27013c415c3a10eac365d19253c4b5a212e7c52

en · unofficial · sha256:9987094ad4f9bd77a19a8b188ed7175d1978f4a7099bfb2e59237a4878487c23

orim12i $p |- ( ( ph \/ ch ) -> ( ps \/ th ) ) $= ( wo orcd olcd jaoi ) ABDGCABDEHCDBFIJ $.

Referencing norms: proof_reference/0c0ddd5a7172473c953c9eea78cf7ada86e583f68dd3ad4f926d25edd983d13d, proof_reference/197e12e8750ee7111429cb8ade1a6b21dac55b16e43e40ed62b7cb91b6ef5d18, proof_reference/51427574677ae19df34eb8f22b7ea08820f6b483595572e840b1bced5e7a1b66, proof_reference/ede38fbd9c58ed999041b0415bdb22b731a8998521d7897e941af1a7110ec690, source_statement/9ce0dd7daa183326fb4b60cf5799dafa2190410807089c9aaea7b819d76b0dab, statement_hypothesis/12541b03c3214301318dc3ebc57822d5e7766be31b660028ad0a8a2f0f5401bb, statement_hypothesis/16d54ad6d11fa2b93a96da3acaaf15f6ee3d595c46c1c39f1d72d55a0c87ba0d, statement_uses/1b749d04d29e45813fe72e60e739edaa672dcb4a49bb840cf074f583c25b6983, statement_uses/2b86d278da26f1c8728edd766d8a7dfff900f08e98616a74d09baf279c3a5a8e, statement_uses/6987f3858017da1e8b33c235b57828fa3982962567f0385aaf21e2bf7e8450ae, statement_uses/852faca8443f49602829abe8fe44e4ee20d8c8c4cda5080a481df4c353d61125, statement_uses/85e547df05a1de81ceb6fd2b5c4b177651a08a00436d5aa0ea9aec8be8f1c048, statement_uses/a3d73ccbbe525e7140e36a3472857e9679369ccc78c63e4360e55af386e0f0d3, statement_uses/c1517d655ea1279495d1ab2366a28142f792caefec6800f447a302a58c1beaa3, statement_uses/e6adb6bdcc5ee0eb257f1c588cc3bd19a64d0ad895fb330c3d93457677cec7d8

en · unofficial · sha256:dd5475e3d5c6cf3f5a50063cb30b9cd547396c9a3007a615fc61973dface84f0

2a1i.1 $e |- ph $.

Referencing norms: source_statement/c8eb620f08df8609646a15c5d0b7468cc1a3792028fea96eb25e7ed2391fa741

en · unofficial · sha256:64e08b223531ddd224dd149faaea5f1774ef0407ece08ebef6bea3f11e5bb825

$( An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise ~ df-sb . (Revised by BJ, 22-Dec-2020.) $)

Referencing norms: statement_comment/1e74c2e04a321e5d4cb87b6cd984ea73be3f17008eda55fa8f8dd97a405083a2

en · unofficial · sha256:dd33f47e31a665a0bceeff664042657f148f5462daae613e686a5cd4a30cd690

3bitr2d.2 $e |- ( ph -> ( th <-> ch ) ) $.

Referencing norms: source_statement/d49a2506a70bed6a148c72bdd8ba79edc7eafb86899293bb64ca97697a841eca

en · unofficial · sha256:4dde5b6586b241f6adc48b68d1e63eeddd33d71cb9994c33498c1538b6bb777f

$( Combine antecedents into a single biconditional. This inference, reminiscent of ~ ja , is reversible: The hypotheses can be deduced from the conclusion alone (see ~ pm5.1im and ~ pm5.21im ). (Contributed by Wolf Lammen, 13-May-2013.) $)

Referencing norms: statement_comment/c4800c30771482ada68a4e768a1d47c60253ba02ceb77bb91d317a634827be0c

en · unofficial · sha256:07f15e880024ae15293b0904621f9ca90188793f61caf4515dab5e4ac2e638c5

sylbird.2 $e |- ( ph -> ( ch -> th ) ) $.

Referencing norms: source_statement/4cef64d939c4a014d6267a05b81b96ad2cf149a8d782533625e089fd711499d6

en · unofficial · sha256:322dbfc9c607d685c2ce0836fc3114590c8212925bc8069318b085fe8975a558

speimfw.2 $e |- ( x = y -> ( ph -> ps ) ) $.

Referencing norms: source_statement/60dfce4c2f6c2c761d667b1d27a8a14e0c0248286e3e91b62ce1f70db9af6a30

en · unofficial · sha256:b2a2b5c47884ebabf1dcc24c11372f6a6038335d300fbd58773169733c65c1ce

neleq12d $p |- ( ph -> ( A e/ C <-> B e/ D ) ) $= ( wcel wn wnel eleq12d notbid df-nel 3bitr4g ) ABDHZICEHZIBDJCEJAOPABCDEF GKLBDMCEMN $.

Referencing norms: proof_reference/17e3b5d07a00831c035f9c753fc4ae86d2bfdae4dfce6fbba3e59a83431e42c2, proof_reference/1cd80123db267bdfc49cbe15e51daeb2c45e8bfbd30c3d3ffad46f2c39bcc895, proof_reference/483971100d90cdea799ae38b130043b699c2f563659c9b6b85de24b9fa33bde2, proof_reference/9a64e0bbdd5f36fb77db2bb649cdf9af8a42bb7dda5891c4879370a9012968d2, proof_reference/b10d58f1c717519cab696e61cefd98db49893326133bdf8c200d8c481d866204, proof_reference/bb0291eedf0f4789205d51b80fe897a3d49ef6151fffddd2f4bb4222a2544903, proof_reference/e0f65ec7bbed7d43492afbeef8cc56badbaaa4389161fd8625019849c549c219, source_statement/a1a0f3ad301ec0f5f28c35ab3dde7fbcb565f60e4141d4ad054fb798ac88225d, statement_hypothesis/9abcef346c29069f4663be27df922e0319d129af6dd4e67547f301cbcb74d392, statement_hypothesis/ea8049dfada02bce6fc71eb12db81a365e1f8473114251aee0ede253352287a0, statement_uses/255c619aee34cce1c81fa262ea58645e80874bcc098c2009e1343ea5be7077ee, statement_uses/34fcc4226eddf61735877ce9565291ddadcb33f46c5c339d58f3adb173617be1, statement_uses/4466e24fbc8abfc494ede8b24f4a3064aa21508ca469a8a43ae0bc44f97c26c6, statement_uses/5c95d7edeffac650d8c37fcc9112d5aadf7c8e7066d964cb19ddda2d191b2389, statement_uses/6e77c2d95b8108207aa0feca3cb9286a8681841769c139194b26680205087814, statement_uses/771851dac0a477ff3d8ef7b76d294a708490cb3e505b6395c6113ccd1d0d04fd, statement_uses/92f3cbf98e1630c9e972cac348fd7126abac322607773199b1bd7be284234981, statement_uses/974e64f53759902b9f7b8bfa7ae3d0add3307d14f327e127cf7672abb7ea1daf, statement_uses/9b6329b4bfab38dd745d77fbc0b550ce5dda2076ca336f3daaaa7d184810ef97, statement_uses/d1cb428875b35f420b09138a2a86b87369dab83c21c35b34e7ccf11ea138faf5

en · unofficial · sha256:29a11decdd4401e22c58cc1f2d32b946f4afd1e6258d6b6ef548189c09d93f59

$( Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)

Referencing norms: statement_comment/9ab65b975946368e13894f0cc3473d76df2174077e3e5909840ed7d0a1c5450c

en · unofficial · sha256:f4191d1ac63b7715911350e0d85cd83642ad2564d8f7bb40efb98c6665677f1f

axextg $p |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $= ( vw wel wb wal weq elequ2 bibi1d albidv equequ1 imbi12d ax-ext chvarvv wi ) CDEZCBEZFZCGZDBHZPCAEZRFZCGZABHZPDADAHZTUDUAUEUFSUCCUFQUBRDACIJKDABL MDBCNO $.

Referencing norms: proof_reference/0a7d9bd6197c24f4af5efe3ab4d8ec6b08d7077ba05e1bb7b3a1cce681b3e1db, proof_reference/125c616b86f84d0c6fea101ea26942d75acd945ffd674468c358b918030ff4f0, proof_reference/150a8eb34c8ec884951a73678c5c4beea3ce4a456b85fceef8937b54f6cd2356, proof_reference/1add3d3c66ef5e431775923a12ad50e5ca9610db265b7b29cb8075d9cbab3f05, proof_reference/1c620df2a57baf413e6809e70a5ecad208cdb7fefde0157d5dd0ac9bb6102024, proof_reference/3be30e7a464b1fa4e8fb2f8afb4921ca955f7619fcbd69e0a32d6e68362e58c9, proof_reference/4dd160d6484ba81ebafc089a7f0af04b9c55d933e0d5bbc5494f10d9231a38f9, proof_reference/72caed714f55de6681b4975aece916fb2bff3c999a21e3907f84445a6e08df16, proof_reference/96c87d44da642e48296b042441b6e18c25f6f8697c9e246649f2b7824c9dc4db, proof_reference/a774b2f362a6f4495c19295e4afe4868ab9b2e05edeba737bc64a03196a0d5d6, proof_reference/c744fb90af17b76b5dae41982b2c13e1e184eb5ee76cdc257d87f4b723c43477, proof_reference/eaddac1d049fd320b3800cf973a94934182faf33620c9abcc8bc3bbff3e83e43, proof_reference/ee4dcfc139a109ab8a4e98b43c416672fd0cf9feb1315c9512d2c6173f9c6068, source_statement/e24ae2183f0aa1b380c22026a89c2ba4af96a86f924d066265219c3b210ad1d0, statement_distinct/56a8f474eb6926638ccd2152f36fec4f82664f87df01baa0081e46242b7161a4, statement_distinct/ac05c5a5f0371461051a707906ed99307c0b2be3818ba3d19f2baf4f1771e9e1, statement_uses/17bf91af9f7348e693c3424177decd6229e1db047a3db464378f7dea6171bcf1, statement_uses/325d2ff8e3de3e5c1378031cc4d18e3c92447c78b340b9a64dc9e88ebabb388a, statement_uses/4ab5e34d229360edd190a3e420fbb17a2c2556255757acce50a124351d959b63, statement_uses/561c37d691c07e2d00be3f361f9d37dd5dd381ea6b50b93f4205ac51d5590db8, statement_uses/582feddb5dffc9d0b19db870b5c6e26972f3ea2740c383bd04c48c335f892aad, statement_uses/7e482a1e6f5bff007b201e8904521e26e5324d98aec3b52e75f69d6ff9c3d854, statement_uses/9ff391003697663eac191201fd1c296e46d58ad0846e6f5fb3d2ed9d62f50260, statement_uses/a1ef5f08487652b3978a35b063ae487a844eeb84053463bb505ecce03d3eaa8b, statement_uses/c7ce99d0a36910fb530a81ac42ffb6da18b657292248c29d12e973b32f39e715, statement_uses/fa73109951a731c08d8588324b720073aa04dad5d1de8b186e3e0eb50f61b910

en · unofficial · sha256:43de90be213f561d41499c63bde2360078ec4f937fbea4e5ed09201cabbf0cdd

syl10.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.

Referencing norms: source_statement/da0fce973d51b341d2a961278a5bb344c003fe8484873499cb0cd09ff2c1e1da

en · unofficial · sha256:d657e54e94b493ae4cf07f9d4e15177ed876f65ee04e7c18fa12002531982f1b

biimprd.1 $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/95f13af1881318e4162e1be4491c7c391128d1dccc2514b4e3fb46c8b6fa5d45

en · unofficial · sha256:76eb08c7f8917965021fb345254ad3f7ed5c379141cc8ed7b195bf512c168c27

min $e |- ph $.

Referencing norms: source_statement/85b8deaabae04c4633f0275783c0d15b84274c2a03b5ad7fafba5cbd340e83a0

en · unofficial · sha256:3d38c2bd95806a46b7f28961de3886347d913f2cd5d95f40c5338a2160c2a320

mpii.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.

Referencing norms: source_statement/f53a49e42940c42b14582b1035f9fb2868648fb472f1d057abf2000443f60740

en · unofficial · sha256:ba98ed0bf517c6f3d0c85641ee8f73a1cd23d2be1129443ca43e71480ad0ebd5

wcel $a wff A e. B $.

Referencing norms: source_statement/ce734e6071546a4b20da99ea89522eeab9a0593764efc7662822583819c08130, statement_uses/3fe135abda8b3f92a3bd71a327f8fb477f6e8415472ef356cacac1d52127fc0a, statement_uses/5d3bd9bfe2812b3c2486d2fe86efb592dbb36df5a1ebb68ca68198226c221565, statement_uses/bbf125f3b1afb7429b901662308f805a8294f44d7a38738d2175ad461cddb5c3, symbol_name/e20ad786f5bad49f60e835c5968bed1b2612181b5b15852d2ae5686262889cba

en · unofficial · sha256:7f4ba0315d8d3c39aed46d8e916506d9d0491ba97b5194d70b15413a29e34a77

mpii.1 $e |- ch $.

Referencing norms: source_statement/e2fedd16f553b1051c9ab8410f83ec7e87d6828ce2fbe216c76bdc51750ca05e

en · unofficial · sha256:3fc1fffdfcfbcca17ec4039ec1661825aafac1579a4d72ba4f4b4de8bd9082e3

$( Biconditional property for substitution. Closed form of ~ sbbii . Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise ~ df-sb . (Revised by BJ, 22-Dec-2020.) $)

Referencing norms: statement_comment/88224d28b6dcc9edd4803823c0ad84faad0929cb6e3181eb3e9435c497244ee7

en · unofficial · sha256:11e08f414202c54c70a8abe0ec1d198c97bf09b8017b7a3976a10a0faab0cd65

$( Obsolete version of ~ nbbn as of 10-Jun-2026. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.) $)

Referencing norms: statement_comment/87ee48cff319189cc39ee996e2ee97b2dadcc8a7f71fee99c1bdaf26db81c097

en · unofficial · sha256:08b334b8de9d32ac61726a601f8427c6a405440e7650af8abc92d8c79129938e

bitru.1 $e |- ph $.

Referencing norms: source_statement/d94647e7e381598568454b7bfe7443229fcba5768fa0bf4a701b498378f7678a

en · unofficial · sha256:b7f1a10fa5c34f2969b267c923a037ecf1026d53e768cbb1f573a899caaa772f

$( Version of ~ equs4v with its consequence simplified by ~ exsimpr . (Contributed by BJ, 9-Nov-2021.) $)

Referencing norms: statement_comment/b69a80b8901e1860aab14caf3fd4132a641d6a888dd08c2b8bb7653ad460fbb6

en · unofficial · sha256:d222084a5dd4af0809582b5b48699a412cf22a252cdb2ca56c0cc9a4e30a78a9

jarri.1 $e |- ( ( ph -> ps ) -> ch ) $.

Referencing norms: source_statement/7de27d3da9c474b1ff2090f50cc8275447c21f346d37edadc2e0dcfe7161e077

en · unofficial · sha256:eeb050d799c17520c48b2ebeebacabf538056cc273d0f9a1f2b76888c50754ed

imbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.

Referencing norms: source_statement/d8060f2af6a03273fa033f6c537cb9a467381a33670fa3b0202302058bca70bb

en · unofficial · sha256:36552770f88e724c0dd32ded1e1842f77d97ed685da87626abbd466208f24da7

$( Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/2c7cdc30c3eb11144dc8b9ab4fd0a5e3e883294f7e32476debb9aa4dd9c5cb5b

en · unofficial · sha256:76a4c1c18978727f53e40d675b32b6ce550c7e38bcb1c534e2ec3561385eab88

$( Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993.) $)

Referencing norms: statement_comment/9970017b93bdf6f3c295cfe65ae8d79521cccf91810eb538edaa74a46c4bc288

en · unofficial · sha256:ac9394d9262e7b617eaa0af82d6a81567d844bda7a64a7f805cb30f423939e48

syldd.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.

Referencing norms: source_statement/179e7c43daec86fea7df1a9e6560fb59d54f3071db7fa0629cd642a4b2355470

en · unofficial · sha256:3b9f5ccf17729c6e2872bd4ee03c6c674a5c280b127586ff184af23087678996

ru $p |- { x | x e/ x } e/ _V $= ( vz vy cv wnel cab cvv wcel wex wel wn wb wal ru0 weq id neleq12d df-nel wceq mtbir bitrdi eqabbw nex isset nelir ) ADZUFEZAFZGUHGHBDZUHSZBIUJBUJC BJCCJKZLCMCBNUGUKACUIACOZUGCDZUMEUKULUFUMUFUMULPZUNQUMUMRUAUBTUCBUHUDTUE $.

Referencing norms: proof_reference/04f71104a3974def7c8cf0aabd68fcce92c27d283595f9f89567254ff151a535, proof_reference/17e8665ec365444c1f328cb6208b355c6f2ab8609898aa7d3de7af79dc4af497, proof_reference/22c907d20c54ad5958901f7829383095d2d7ad1364ab5328045a3c212746bec8, proof_reference/3138fc56ecaef8df546dc22a675adc705b99f1f1fd93529c5da924337761410c, proof_reference/3b5bb652497b895e33f488577db9398a439dc4b81bcf1c9f944457f7ab6459b3, proof_reference/40a4cb142efb5ccde2c53dad50661b1c84f59b3c9098cab9278d2097d0a58865, proof_reference/45086f2d8efa0310c82be6fd02c9df6de740b5f10ac233fd9466c5b49b89ec23, proof_reference/559ba448f7b529dd877ce318c8a02e465c819f44547061ffadb17976022fe01b, proof_reference/74e3af61f384a8ba292aac9cac06f6aadf1dbda37308fe35135feb5506fb7f35, proof_reference/9155599c5e14a6144e319a26ef5fdfd2da558100c0e66c7272aca447d1635835, proof_reference/99a6406c117ae69a6af7efed3f3daa98a7ae27ae97ddf3385d91883512c8e45f, proof_reference/aa28b72685bb9da6669598e731dbdd0bb1da68d7d17746b8f2fafd567018c4a1, proof_reference/b289b974c3ff22427a846bf1e40488cb2e48402cb2f4ff0a74331ae4e62c760b, proof_reference/b4dffd61b72c1e2b4163203d83444756ff7fb1df3ca85ec546b9f66a71b29e9f, proof_reference/b90fbb0edff8e0323e27cf069772966ba60860feaa100b79de234110e6d3f46a, proof_reference/bc9602b1022375b57ca8108d00b905a0294a079c132b95f092cbe0d9e65de9b2, proof_reference/bf1d1c34bccb28034e8552848addee5582561ac6d641dac9f6240e7e111c26e1, proof_reference/c60912f65dc4aa1e74e220616fb42b876ba31fefd930ffe864de50cfaf62b1fe, proof_reference/d16dd8b45ba32e467c16cb83b9b4d2462b1b64ea944c27ba7020c0526dab1336, proof_reference/d52acffc2c6d6edbe11796c5aec28159237681bf7eec439a8e6e40fe49aaae37, proof_reference/d9df1d02d6864d61c4c70c868a0d9fd514d2475ee4a0564ce2804cbebcd4dbcf, proof_reference/ed6293159c3060294a133d6c7df60aae05f7078ff11faeb989989b9ee2635334, proof_reference/f00957b19d64cf568975820b6118515fbccbf66c518eac114b25f3d8ea3f6f45, proof_reference/f9eda26d98d0a3f2c621be75a650ba45eb8f1ab3928435ad57710b3326fb6c07, source_statement/e1fef109a03d4d6075d8cacae52f2fdd2091fac25bc5646c5805e3dc49361c24, statement_uses/06dabb21f7e65d17cae0411482651221dadd33b81edc453eb92877411f35d35f, statement_uses/168cb31c6b730588746505b0e0302eb036dd16e16abaadcea1cd2fe9ee423485, statement_uses/32688b4a0ed1bec4ce87b4fb219524e1aae1092aa71a2737ecde97042937143c, statement_uses/6bf923f3a08ec0acc121c20fadaa8fcd01127da9560b8a99340eaf827a37c51d, statement_uses/c712fd9038b70819b36f6c3a81f4dfd00f922bb0c6787fc1eebfe9e86e215304, statement_uses/f0a036c5eb3355de43ce5eb430426dfa884d7960f58f24de840bbf6cbf6e484e

en · unofficial · sha256:29c5c2f5e4f6691b69910e55ab0f1f0bb2dede243be6e70efe8c4796c2b1ba15

anassrs $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $= ( exp32 imp31 ) ABCDABCDEFG $.

Referencing norms: proof_reference/0acaa46df4743a843a028f30840d37ad52dbc543ee324de0774934cf33d94167, proof_reference/18a651800e1a1d6796710b2f879339d6944ccb13f6c91c539643ca72e49cd9f4, source_statement/219e1c36a7df5a694d45f3cc99662449097f76f5084105dab873c05803f86d80, statement_hypothesis/5aa139c6cf91c15db7050a031bdc8173632fbb056a2809912e9908b077d18e04, statement_uses/156837fcd16046569eef16cbecb886b5b51fe638860854faa0844dd1596de06d, statement_uses/2d4f1e319b9ef6b7e952677cdac06d1eec109a7bf69db7e205fe9703c690afbe, statement_uses/324b85e8e1822fdeeaaeac62fc44e5fb818c04c513d804fb2ee2abd9245d155a, statement_uses/412b06fe2df936740a484c7f5ecac9972d59796e34537bb56ff539a9d47e2d61, statement_uses/45afacf45c5b125f740b8a95604e238cc14c5bce299bae56adfbddeb91e32543, statement_uses/76385a3866630f517e2816ba25d5526ef142623080228f638cd59bad6a69302a, statement_uses/b98f95f9b6fc996439587f1a07562d36a74b07c30caf507ffb26fad5ee755e44, statement_uses/ffb4ecb7673c60bec3152c57b6e2d89322fd93c1c60c7318aec831d444f71ba3

en · unofficial · sha256:889faec1318732325766da3ada47afc4c085841942661f36f09594867ba374ce

impbid21d.2 $e |- ( ph -> ( th -> ch ) ) $.

Referencing norms: source_statement/8cbcf82e55585f71893cd7ebced51044731ee2791a50e0fb02e493a85626dbef

en · unofficial · sha256:adf3aa8ec4b6646b7fd6b0f5e0a32c248e7119cfb40fef2921ee5615710cbefa

sbievw2.2 $e |- ( w = y -> ( ch <-> ps ) ) $.

Referencing norms: source_statement/4e5907911dafb2602628f7cb568a788c2eb77ec29138c7c9bfd9740a7132a5d9

en · unofficial · sha256:c095579f0b4bb5755641324dddbee729d53ee3c945eb84d5cfd8e34b4f6b3a4b

$( Version of ~ 19.8a with a disjoint variable condition, requiring fewer axioms. Converse of ~ ax5e . (Contributed by BJ, 12-Mar-2020.) $)

Referencing norms: statement_comment/e393c02cceb01a34275d9595b98e749bd22ebf632f30c37733ed228f6ecf1bad

en · unofficial · sha256:2f5fe6dbd76abffa246a1f264cb9bf62fdbbc9c1fe8448686af3ad75a78a1313

$( A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) $)

Referencing norms: statement_comment/53c62bb5a699eee9a302558fe4b531781d51911837d0794c193b6e133796e487

en · unofficial · sha256:0d35677b36b6d31107df3ff7e6e1c095f93840808309df0e5597561e0d0013e0

sylg $p |- ( ph -> A. x ch ) $= ( wal alimi syl ) ABDGCDGEBCDFHI $.

Referencing norms: proof_reference/31951a6e4bcb670b6194a37a380c237637e4e5fa41f8421a07ab12f01770d9ac, proof_reference/bec7954f3871d7aaf3d55d5fa073fe33d2486be9852a3e2a74a7e7da7aad787f, proof_reference/fc8102f53abe5d5c409f79ec769f52001bfd934da04b5b1afbf7efdaaf90f0d7, source_statement/1579c85a6b0ce7f3c8be13b60ea965a4ff433570139e5cf7b8141fe44ae194ea, statement_hypothesis/798668936ddd1920435829fcbefe303f0ac66bbb842c714ba2ce7d5eb91a1ca3, statement_hypothesis/836827146a7f775b0fdd218f8d052bd373fec333df8faf69e9a25151e3a5e6ea, statement_uses/2f62400380268c5aae082fa0f73a21bd796224599007f5d6272fdfb497e1f552, statement_uses/35aed4bc1c62c3195a54f5f4d30851600db6d7c92f8fe3ed24aee8d16986d45d, statement_uses/38e6bae21c3001be39f20130e28433bb1faf935f21ec4935113e586f20458895, statement_uses/50fc270d7cfc2550096d6d39e7832ff5f6941a89fbb1cccd7a596b8407b89756, statement_uses/79fc737ad2ac40457aa8a1c3929b4738e7d2b2fa6c68727e890dfe74961daf07, statement_uses/c6d4c4fcfcb8422a08e4299c9b60af71cf1c7adc47d9e7bf84642a0f157819ba, statement_uses/fa8db020ee2c3edc3e8a13ca30a53f9c58f07b98ee1fa31c7e1f773ef803ce0e

en · unofficial · sha256:052e5b9c2be5c4b6d40c59fe17df1b217fd6cdc2223ea31a906a997da2d8d48b

mp2d.1 $e |- ( ph -> ps ) $.

Referencing norms: source_statement/d4d088afcf48f58ad658cbd382628f71328a51b16f65061f584ee10104468c5e

en · unofficial · sha256:f15a0938010d0bc98dea90dd84750ae33b06a9f6531b520b25bef05f5f657d5e

$( Version of ~ 19.9 with a disjoint variable condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also ~ 19.3v . (Contributed by NM, 28-May-1995.) Remove dependency on ~ ax-7 . (Revised by Wolf Lammen, 4-Dec-2017.) $)

Referencing norms: statement_comment/144df77d096e261d00ccbbc1d51ba14c211451d565b4e4a63ae779fdce70e8f0

en · unofficial · sha256:a928bc2a12dd4ec79bde494328566d73a30c9e70848734ef830f1cac3077e537

$( A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) $)

Referencing norms: statement_comment/ecf617c3e54dda48f9a9514e012644948e6c72dc0467e6fc494c7f36ea9f249d

en · unofficial · sha256:d754538c44b6f7672ba302b891e3f7f9e254d7f6ca31d0ab26c00d5bea094c02

$( One direction of the equivalence in ~ exsb is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.) $)

Referencing norms: statement_comment/1c28bb789878490c348e7db6b88334b0c9ff9bd59230316faab709bd92e4b8ae

en · unofficial · sha256:a9f0f426ba5f984afc9bbb1e0a3f73558d9abd1bca6e0064187c6904de78a675

eqeq2i $p |- ( C = A <-> C = B ) $= ( wceq wb eqeq2 ax-mp ) ABECAECBEFDABCGH $.

Referencing norms: proof_reference/673968e5055922d29685236db2e743f4fc320769e6a4fb3742b457ce720565ee, proof_reference/9e8eceeade2b5f360f8ee67b58371d7d7002d69d9b21b7d1451b38ac2c1b83b9, proof_reference/ebef0e48fef520bcd10c08f4b88a6454579bda0657bcbfaef0e4528e4ef99280, proof_reference/ff539d687aea0cdcca72bee1fc3b2fcc31aca7024a6781180584115bb87bd92b, source_statement/d223d4f450458a72d713f5c80735146d0ead66baeb5da810951369d426efe5d7, statement_hypothesis/14651267f5579c4abd43a1cf728cf8dae1dc54579ff472e9a28f88130cad4f00, statement_uses/4119f1f6e858a7210765804dee1c25e2e0d8a8b66e2bd371e9cd6b2d9198f676, statement_uses/644e84a08679f7beed5244222acd535bd1afa30ee133e0cf1a855834842ccce9, statement_uses/9caa8304057205a832e850eb87579c33e0c4d2f856b04119bb23745caf4a305c, statement_uses/ab4e31d26f6fcda079fc1f5fff31ab68acd1063e8f1783ba58e7b4a3a06422d2, statement_uses/b2aad42865e4882181fa3aeaeac0925b06b9ae3b6b4acb0e235e3ae9284f6403, statement_uses/c6849512d1559b18ace000ec7623dbe684c0b804e93dc81d1a8a0cd68989c7b8, statement_uses/e4cc6945aaf8c1a7fbc70cb86cbc23f0081899df27c63bed74325dbfa0adbfb4

en · unofficial · sha256:43d5174b0916dd291cd8b3acc6b1896226b3ff48e0ae14e34d44a0212b877dca

eleqtrri $p |- A e. C $= ( eqcomi eleqtri ) ABCDCBEFG $.

Referencing norms: proof_reference/860a20fc9b33a71466058a390b38fddbf7dcf8e9228fcaa5ba799f11bd757ab6, proof_reference/b10f470a0569d4c25c6e5b4c811d643ae30dae464959d7151700f34fb334c004, source_statement/371fa98b8743dbd05cc3a4fdcbda39fc3be0443cb659dd80dca10b5f0472c35f, statement_hypothesis/8678d41f78d151ff8316fb08313092553f97bb531bf87cc5dfc9db1cad208e73, statement_hypothesis/8986b54c9edcaad537fa11d9394f735227c3cb1516fe15dca846cfab70f103b7, statement_uses/0b9a7b4a313bf9c1ca879a54dd2028abccff1f784afe49a98c31c33f6cffc945, statement_uses/1cbc0e9aab3a70c0d78b9485fa1ebd3efd57efdfb157a1e0ec7708ba1c241393, statement_uses/3cb110ec75cc72e8fd7b79730fc644b2f245cc642b1f6c87ed962e6eed835bbb

en · unofficial · sha256:5d5f24b077bb8d2cbbd77d58311ac88508ceddf33869daa640b54eaa48662c58

mt2i.2 $e |- ( ph -> ( ps -> -. ch ) ) $.

Referencing norms: source_statement/d835c8834a7f941888e9ed3c46b0b3b7520f0ea809bb7658544f1ee9400ae4b9

en · unofficial · sha256:05109d1e06fa2f8784b8dc249ef60afef28c3048b9ac3a54e6ed3b9ef8cf9bf8

$( Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.) $)

Referencing norms: statement_comment/0eafc66c705a9db6ebf016d2177c3971918b6fe2a7d14d26369c9ec4d7395a5b

en · unofficial · sha256:78e3299f7a398f9b6dc08a9e19c7fb7d87013b369c3a62bcb5b0b88a13c0dfaf

$( Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise ~ df-sb . (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) $)

Referencing norms: statement_comment/9ee5e820f032f4f83d871b9e17ab359361327e236f1919d2d823f479caa28b47

en · unofficial · sha256:63bb6c940827b7fcfc19fe1766bc4c92bd0653bf4725636f2951a68b4a6a8165

vu $f setvar u $.

Referencing norms: source_statement/d90cf19f827cdfdda923b71d9114a749a38d87edb8914375af62305a475370c8

en · unofficial · sha256:4b006c1cbf3b4d08055e2151958defe582fa1d88901b6daf5ef2ee26c2ba9773

$( Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/d3e1571d8f53acda410568f5c04dfdb88adbed5ba8ab08a931a350f6429da5d1

en · unofficial · sha256:8a20b639dd143f22c9894c5105ffd174c666ee4bc8adc1c1e26c8cf4b2e5d0b1

$( Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) $)

Referencing norms: statement_comment/e7dbf5e1ec7dc4730ea5e30dcc948af360c81a7196e36e07f9c3531cd62f46d0

en · unofficial · sha256:2e89300fcb926d13b36d6988a2eb512e8f5e6e33554560f047cde64d65ecb8cc

expcom $p |- ( ps -> ( ph -> ch ) ) $= ( ex com12 ) ABCABCDEF $.

Referencing norms: proof_reference/1c4b83329d73487c207ab068acde2c88ed39b0a0245170643ced0c1e78704a4d, proof_reference/4189cf947378a882a2839f6f863c75623f6b203d61d18f5adfdd7b2311086c1b, source_statement/0692d34411e81cb888713f71a94d0244026287c0ca512ffcc01fb8f6d129785a, statement_hypothesis/0d593e24ea613b1336ba3bc3d9c63cd021752d7da021c7d1ce7ada33cbb6859b, statement_uses/5646d63b643a0b43847ed5308208dce5faa4b048c7df80e7035993ed4052cde8, statement_uses/b298c189b09df1aa0141eef30ef1bbe863b252adf01492d64d04b4e1840d265e, statement_uses/c6b378dbd4affc1995cd37f6b12f5524ea0445efae9e01842ffbd4c0268f1086, statement_uses/c9f40cbaa0bb24a32c2ad89674f0de2d5d6541a9021f2dad1ab92f6967dc2832, statement_uses/f51b85be0c8e68e708ee8fe8daa329e639315de3d1f4e4143af9c28966127c35, statement_uses/f652c6d9431cfa096b1ebbb125e48b5b598da63d93606580b695db522311feb4

en · unofficial · sha256:6a66d0727af1066361837ef50ca331fd86232cce27595fa904c3d9aba9738a37

a1ii.1 $e |- ph $.

Referencing norms: source_statement/52283341bd2147ed5089e7a487171851ce069a48ead50f74b6e8c0bf1ebfeef4

en · unofficial · sha256:de9415ca9cea7b7255cf54e0046544cc0b46003b1c8ca7ed3ee566888815983f

$( Proof of ~ ax-9 from ~ ax9v1 and ~ ax9v2 , proving sufficiency of the conjunction of the latter two weakened versions of ~ ax9v , which is itself a weakened version of ~ ax-9 . (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.) $)

Referencing norms: statement_comment/e6ef9b00131374589677066a6f2facb49aff6b3ea933a5f2ecb215fb4a7d698a

en · unofficial · sha256:1703c3185e4b7c4bb4782fc2135ced08c94e41cede66b4fed44a83066085a29d

sylbb $p |- ( ph -> ch ) $= ( biimpi sylbi ) ABCDBCEFG $.

Referencing norms: proof_reference/74daa548bc21c7f76ae8f8b43cabe9489bf0cf2716dee9ae0f4aaf1c490a8f5a, proof_reference/be4408a80642ec2567b6c238bafcffe4458f1da0f248e307eb2b05ba7d4dd2b6, source_statement/50eccf7104973567256d96a7cee99419c7b30cf9f82e1fd10ce18b2d5d61e79b, statement_hypothesis/c5b5c3d9f408809d60e58e7a752b5f8d4e1093aa8634d3b669ad1d5d57a41e2e, statement_hypothesis/e99ce7ea0250d26c6479516feb5788c749d5b85b69846db5c8d05f9142d59ee7, statement_uses/0478eb036dc0c8cb65855efd58ad16723c59e3e961067ce20480bfe4eebef94c, statement_uses/0877a31f432db37bcaf474b4f3e36c120aa37ea83277ee29e4ff1f4263d412ae, statement_uses/5713331ebfeab6ac90e1fd862eedc0b5f9bedc7d14554b4f398d1ff690d83152, statement_uses/6c746a69c32f080c33ce74da9348f12ccedc3c3dd651af15f79bebfeaf6b8529, statement_uses/7a66a96ac8bc975b4143ec2161985e536babd6d8442695b9ba61df2eb7482a20

en · unofficial · sha256:5ce6c648d81d44efcc6014b10023cc413668a0e8ac803247c8bb8106b220c49b

$( Commutation of antecedents. Swap 3rd and 4th. Deduction associated with ~ com23 . Double deduction associated with ~ com12 . (Contributed by NM, 25-Apr-1994.) $)

Referencing norms: statement_comment/cd4b44d10686bf6e7c1ea73ee1f1fc6777bb1ad128585069bb8bbd35fb01a496

en · unofficial · sha256:82d791638ee7ce2150c36d16ecb30c08efe3bac553a75e79001665aaf363e35b

syl8ib.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.

Referencing norms: source_statement/c9aa666d6402db3d6fee8705103133aa1521386e329c355e95ecf04748e60024

en · unofficial · sha256:7188448336c057e7b9bdb362ad998299750c0005d318bada1132e022c73c5c65

equcoms $p |- ( y = x -> ph ) $= ( weq equcomi syl ) CBEBCEACBFDG $.

Referencing norms: proof_reference/03345e6b2970f42f8f69bf55c57d09711748bac2667e0df0f94f576e1f48d363, proof_reference/a11ce0e1858ccaef1e3316234c28a517dce5b81a52fc5ea274b2392feef50eb6, proof_reference/bf96ad7b39c49116aedd149f7a8c283cd11913f198221579b62e9dda923f7e4f, source_statement/d9a6fab57365c272bb356a71c442f1cefde570c4539d163a4425b2e4bba2e434, statement_hypothesis/3a6ccd9c653acdd070b26bb71d38ff385164acc497680e37edda3ce29d0acac7, statement_uses/2787e6685acb5909bb8ce6243b737a96839447e5aa584a1425f56d6c3bbcd100, statement_uses/440f835ac1812d8924dbfc2d5b044b09b4a770e79fffb0170a05573816ad1e30, statement_uses/45dc7c7434488685764e602faf53d2aca165c8123781b64be47561b7156b05e0, statement_uses/5c56007a1a3b82d09f077420673fb1095fd41e29c393e422c7a93c6f681def46, statement_uses/9be8c69422ae0fb8c0eb4a6e607efb4e8cd7957533fde5e0f98b7fe552853cdb, statement_uses/cf554abf6d1addd141216d70dcc69dc2eb7c607a68f39a8479b9921a37f64ed6, statement_uses/d573182b9fda1e00fb88d69a904ed4f292dd84f34fa6fda8ffdd63126ffc7853

en · unofficial · sha256:6021cfe97928d3cb48b1a45229d6ceb74653787f06578c968e8395a6e7db434f

orcom $p |- ( ( ph \/ ps ) <-> ( ps \/ ph ) ) $= ( wo pm1.4 impbii ) ABCBACABDBADE $.

Referencing norms: proof_reference/8780d9009f11b1cdc04374a2a2683c9a7db689a02ed97665ad6d09f133bcc055, proof_reference/978c41693d0e1f45434ed16fa947905158655699c0a032b6febf543c2889551a, proof_reference/d2c9e660da79fb6b4e97c34a58ea3cbb8d6733dfbeb808c59fb141e1dd4195ba, source_statement/202f202efda064bb96d4de410b03be268057867d45fc8c4ab07f215df8a778ed, statement_uses/48951308ea529c94ac9b3296303d5d0357c9d631ea5f7cab76b6ca921d67c6a4, statement_uses/80f6c25dd411bd4156f4fed1eceaf4531e76e62811ed5f6f3a6b21d47695c92a, statement_uses/861ffa9019d85b2ddfc8d66b5f2630fc665b1bd7e60d2a533f138ef472c4b7bb, statement_uses/9430678418e905e44111405dfb582f88adcf44c5a8b535cb9057d0b8e7573004, statement_uses/a04d2455363a23dec382bd446b08306d40378999cb397ca2c51e2664d3202e03, statement_uses/ae8cb5b9608320d175b4437491bfa8151ef3bf38750b8dbb4b28bb7b41936575