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Type | Label | Description |
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Statement | ||
Theorem | 3an6 1301 | Analog of an4 576 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | 3or6 1302 | Analog of or4 761 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | mp3an1 1303 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an2 1304 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an3 1305 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an12 1306 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
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Theorem | mp3an13 1307 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an23 1308 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an1i 1309 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
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Theorem | mp3anl1 1310 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl2 1311 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl3 1312 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anr1 1313 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
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Theorem | mp3anr2 1314 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
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Theorem | mp3anr3 1315 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
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Theorem | mp3an 1316 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
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Theorem | mpd3an3 1317 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
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Theorem | mpd3an23 1318 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
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Theorem | mp3and 1319 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | mp3an12i 1320 | mp3an 1316 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 1321 | mp3an 1316 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 1322 | mp3an 1316 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 1323 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
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Theorem | biimp3a 1324 | Infer implication from a logical equivalence. Similar to biimpa 294. (Contributed by NM, 4-Sep-2005.) |
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Theorem | biimp3ar 1325 | Infer implication from a logical equivalence. Similar to biimpar 295. (Contributed by NM, 2-Jan-2009.) |
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Theorem | 3anandis 1326 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
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Theorem | 3anandirs 1327 | 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 1328 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
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Theorem | ecase23d 1329 | Variation of ecased 1328 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 isn't ordinarily part of propositional calculus, the universal
quantifier | ||
Syntax | wal 1330 |
Extend wff definition to include the universal quantifier ('for all').
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Even though it isn't ordinarily part of propositional calculus, the equality
predicate | ||
Syntax | cv 1331 |
This syntax construction states that a variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() While it is tempting and perhaps occasionally useful to view cv 1331 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1331 is intrinsically no different from any other class-building syntax such as cab 2126, cun 3074, or c0 3368. For a general discussion of the theory of classes and the role of cv 1331, see https://us.metamath.org/mpeuni/mmset.html#class 1331.
(The description above applies to set theory, not predicate calculus.
The purpose of introducing |
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Syntax | wceq 1332 |
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 1333 |
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Theorem | trujust 1334 | Soundness justification theorem for df-tru 1335. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
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Definition | df-tru 1335 |
Definition of the truth value "true", or "verum", denoted
by ![]() ![]() |
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Theorem | tru 1336 |
The truth value ![]() |
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Syntax | wfal 1337 |
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Definition | df-fal 1338 |
Definition of the truth value "false", or "falsum", denoted
by ![]() |
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Theorem | fal 1339 |
The truth value ![]() |
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Theorem | dftru2 1340 | 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 1341 |
Eliminate ![]() ![]() |
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Theorem | tbtru 1342 |
A proposition is equivalent to itself being equivalent to ![]() |
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Theorem | nbfal 1343 |
The negation of a proposition is equivalent to itself being equivalent to
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Theorem | bitru 1344 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | bifal 1345 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | falim 1346 |
The truth value ![]() |
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Theorem | falimd 1347 |
The truth value ![]() |
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Theorem | a1tru 1348 |
Anything implies ![]() |
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Theorem | truan 1349 | 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 1350 |
Given falsum, we can define the negation of a wff ![]() ![]() |
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Theorem | inegd 1351 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | pm2.21fal 1352 | 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 1353 | 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 1354 | Extend wff definition to include exclusive disjunction ('xor'). |
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Definition | df-xor 1355 |
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 1356 | One way of defining exclusive or. Equivalent to df-xor 1355. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
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Theorem | excxor 1357 | 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 1358 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
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Theorem | xorbi2d 1359 | 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 1360 | 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 1361 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
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Theorem | xorbi12i 1362 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
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Theorem | xorbin 1363 | 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 1364 | One direction of pm5.18dc 869, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xornbi 1365 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1370. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xor3dc 1366 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
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Theorem | xorcom 1367 |
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Theorem | pm5.15dc 1368 | 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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Theorem | xor2dc 1369 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
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Theorem | xornbidc 1370 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
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Theorem | xordc 1371 | Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | xordc1 1372 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
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Theorem | nbbndc 1373 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
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Theorem | biassdc 1374 |
Associative law for the biconditional, for decidable propositions.
The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.) |
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Theorem | bilukdc 1375 | Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | dfbi3dc 1376 | An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | pm5.24dc 1377 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | xordidc 1378 | Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.) |
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Theorem | anxordi 1379 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
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For classical logic, truth tables can be used to define propositional
logic operations, by showing the results of those operations for all
possible combinations of true (
Although the intuitionistic logic connectives are not as simply defined,
Here we show that our definitions and axioms produce equivalent results for
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Theorem | truantru 1380 |
A ![]() |
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Theorem | truanfal 1381 |
A ![]() |
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Theorem | falantru 1382 |
A ![]() |
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Theorem | falanfal 1383 |
A ![]() |
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Theorem | truortru 1384 |
A ![]() |
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Theorem | truorfal 1385 |
A ![]() |
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Theorem | falortru 1386 |
A ![]() |
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Theorem | falorfal 1387 |
A ![]() |
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Theorem | truimtru 1388 |
A ![]() |
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Theorem | truimfal 1389 |
A ![]() |
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Theorem | falimtru 1390 |
A ![]() |
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Theorem | falimfal 1391 |
A ![]() |
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Theorem | nottru 1392 |
A ![]() |
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Theorem | notfal 1393 |
A ![]() |
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Theorem | trubitru 1394 |
A ![]() |
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Theorem | trubifal 1395 |
A ![]() |
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Theorem | falbitru 1396 |
A ![]() |
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Theorem | falbifal 1397 |
A ![]() |
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Theorem | truxortru 1398 |
A ![]() |
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Theorem | truxorfal 1399 |
A ![]() |
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Theorem | falxortru 1400 |
A ![]() |
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