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Type | Label | Description |
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Statement | ||
Theorem | syl3anr1 1301 | A syllogism inference. (Contributed by NM, 31-Jul-2007.) |
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Theorem | syl3anr2 1302 | A syllogism inference. (Contributed by NM, 1-Aug-2007.) |
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Theorem | syl3anr3 1303 | A syllogism inference. (Contributed by NM, 23-Aug-2007.) |
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Theorem | 3impdi 1304 | Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.) |
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Theorem | 3impdir 1305 | Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.) |
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Theorem | 3anidm12 1306 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
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Theorem | 3anidm13 1307 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
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Theorem | 3anidm23 1308 | Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) |
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Theorem | syl2an3an 1309 | syl3an 1291 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
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Theorem | syl2an23an 1310 | Deduction related to syl3an 1291 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
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Theorem | 3ori 1311 | Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.) |
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Theorem | 3jao 1312 | Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) |
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Theorem | 3jaob 1313 | Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) |
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Theorem | 3jaoi 1314 | Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.) |
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Theorem | 3jaod 1315 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
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Theorem | 3jaoian 1316 | Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.) |
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Theorem | 3jaodan 1317 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
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Theorem | mpjao3dan 1318 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
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Theorem | 3jaao 1319 | Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
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Theorem | 3ianorr 1320 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
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Theorem | syl3an9b 1321 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
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Theorem | 3orbi123d 1322 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
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Theorem | 3anbi123d 1323 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
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Theorem | 3anbi12d 1324 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi13d 1325 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi23d 1326 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi1d 1327 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi2d 1328 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi3d 1329 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anim123d 1330 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
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Theorem | 3orim123d 1331 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
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Theorem | an6 1332 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
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Theorem | 3an6 1333 | Analog of an4 586 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | 3or6 1334 | Analog of or4 772 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | mp3an1 1335 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an2 1336 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an3 1337 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an12 1338 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
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Theorem | mp3an13 1339 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an23 1340 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an1i 1341 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
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Theorem | mp3anl1 1342 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl2 1343 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl3 1344 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anr1 1345 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
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Theorem | mp3anr2 1346 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
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Theorem | mp3anr3 1347 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
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Theorem | mp3an 1348 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
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Theorem | mpd3an3 1349 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
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Theorem | mpd3an23 1350 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
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Theorem | mp3and 1351 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | mp3an12i 1352 | mp3an 1348 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an2i 1353 | mp3an 1348 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an3an 1354 | mp3an 1348 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an2ani 1355 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
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Theorem | biimp3a 1356 | Infer implication from a logical equivalence. Similar to biimpa 296. (Contributed by NM, 4-Sep-2005.) |
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Theorem | biimp3ar 1357 | Infer implication from a logical equivalence. Similar to biimpar 297. (Contributed by NM, 2-Jan-2009.) |
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Theorem | 3anandis 1358 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
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Theorem | 3anandirs 1359 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
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Theorem | ecased 1360 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
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Theorem | ecase23d 1361 | Variation of ecased 1360 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
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Even though it is not ordinarily part of propositional calculus, the
universal quantifier | ||
Syntax | wal 1362 |
Extend wff definition to include the universal quantifier ("for
all").
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Even though it is not ordinarily part of propositional calculus, the equality
predicate | ||
Syntax | cv 1363 |
This syntax construction states that a variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() While it is tempting and perhaps occasionally useful to view cv 1363 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1363 is intrinsically no different from any other class-building syntax such as cab 2175, cun 3142, or c0 3437. For a general discussion of the theory of classes and the role of cv 1363, see https://us.metamath.org/mpeuni/mmset.html#class 1363.
(The description above applies to set theory, not predicate calculus.
The purpose of introducing |
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Syntax | wceq 1364 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class.
(The purpose of introducing |
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Syntax | wtru 1365 |
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Theorem | trujust 1366 | Soundness justification theorem for df-tru 1367. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
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Definition | df-tru 1367 |
Definition of the truth value "true", or "verum", denoted
by ![]() ![]() |
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Theorem | tru 1368 |
The truth value ![]() |
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Syntax | wfal 1369 |
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Definition | df-fal 1370 |
Definition of the truth value "false", or "falsum", denoted
by ![]() |
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Theorem | fal 1371 |
The truth value ![]() |
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Theorem | dftru2 1372 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
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Theorem | mptru 1373 |
Eliminate ![]() ![]() |
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Theorem | tbtru 1374 |
A proposition is equivalent to itself being equivalent to ![]() |
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Theorem | nbfal 1375 |
The negation of a proposition is equivalent to itself being equivalent to
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Theorem | bitru 1376 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | bifal 1377 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | falim 1378 |
The truth value ![]() |
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Theorem | falimd 1379 |
The truth value ![]() |
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Theorem | trud 1380 |
Anything implies ![]() |
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Theorem | truan 1381 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
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Theorem | dfnot 1382 |
Given falsum, we can define the negation of a wff ![]() ![]() |
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Theorem | inegd 1383 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | pm2.21fal 1384 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | pclem6 1385 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
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Syntax | wxo 1386 | Extend wff definition to include exclusive disjunction ('xor'). |
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Definition | df-xor 1387 |
Define exclusive disjunction (logical 'xor'). Return true if either the
left or right, but not both, are true. Contrast with ![]() ![]() ![]() |
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Theorem | xoranor 1388 | One way of defining exclusive or. Equivalent to df-xor 1387. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
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Theorem | excxor 1389 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
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Theorem | xoror 1390 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
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Theorem | xorbi2d 1391 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
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Theorem | xorbi1d 1392 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
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Theorem | xorbi12d 1393 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
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Theorem | xorbi12i 1394 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
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Theorem | xorbin 1395 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
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Theorem | pm5.18im 1396 | One direction of pm5.18dc 884, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xornbi 1397 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1402. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xor3dc 1398 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
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Theorem | xorcom 1399 |
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Theorem | pm5.15dc 1400 | A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
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