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Type | Label | Description |
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Statement | ||
Theorem | mpjao3dan 1201 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
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Theorem | 3jaao 1202 | 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 1203 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
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Theorem | syl3an9b 1204 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
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Theorem | 3orbi123d 1205 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
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Theorem | 3anbi123d 1206 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
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Theorem | 3anbi12d 1207 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi13d 1208 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi23d 1209 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi1d 1210 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi2d 1211 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi3d 1212 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anim123d 1213 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
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Theorem | 3orim123d 1214 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
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Theorem | an6 1215 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
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Theorem | 3an6 1216 | Analog of an4 520 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | 3or6 1217 | Analog of or4 687 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | mp3an1 1218 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an2 1219 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an3 1220 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an12 1221 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
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Theorem | mp3an13 1222 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an23 1223 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an1i 1224 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
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Theorem | mp3anl1 1225 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl2 1226 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl3 1227 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anr1 1228 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
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Theorem | mp3anr2 1229 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
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Theorem | mp3anr3 1230 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
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Theorem | mp3an 1231 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
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Theorem | mpd3an3 1232 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
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Theorem | mpd3an23 1233 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
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Theorem | biimp3a 1234 | Infer implication from a logical equivalence. Similar to biimpa 280. (Contributed by NM, 4-Sep-2005.) |
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Theorem | biimp3ar 1235 | Infer implication from a logical equivalence. Similar to biimpar 281. (Contributed by NM, 2-Jan-2009.) |
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Theorem | 3anandis 1236 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
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Theorem | 3anandirs 1237 | 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 1238 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
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Theorem | ecase23d 1239 | Variation of ecased 1238 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 1240 |
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 1241 |
This syntax construction states that a variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() While it is tempting and perhaps occasionally useful to view cv 1241 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1241 is intrinsically no different from any other class-building syntax such as cab 2023, cun 2909, or c0 3218. For a general discussion of the theory of classes and the role of cv 1241, see http://us.metamath.org/mpeuni/mmset.html#class.
(The description above applies to set theory, not predicate calculus.
The purpose of introducing |
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Syntax | wceq 1242 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.
(The purpose of introducing |
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Syntax | wtru 1243 |
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Theorem | trujust 1244 | Soundness justification theorem for df-tru 1245. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
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Definition | df-tru 1245 |
Definition of the truth value "true", or "verum", denoted
by ![]() ![]() |
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Theorem | tru 1246 |
The truth value ![]() |
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Syntax | wfal 1247 |
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Definition | df-fal 1248 |
Definition of the truth value "false", or "falsum", denoted
by ![]() |
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Theorem | fal 1249 |
The truth value ![]() |
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Theorem | dftru2 1250 | 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 | trud 1251 |
Eliminate ![]() ![]() |
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Theorem | tbtru 1252 |
A proposition is equivalent to itself being equivalent to ![]() |
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Theorem | nbfal 1253 |
The negation of a proposition is equivalent to itself being equivalent to
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Theorem | bitru 1254 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | bifal 1255 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | falim 1256 |
The truth value ![]() |
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Theorem | falimd 1257 |
The truth value ![]() |
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Theorem | a1tru 1258 |
Anything implies ![]() |
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Theorem | truan 1259 | 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 | truanOLD 1260 | Obsolete proof of truan 1259 as of 21-Jul-2019. (Contributed by FL, 20-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | dfnot 1261 |
Given falsum, we can define the negation of a wff ![]() ![]() |
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Theorem | inegd 1262 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | pm2.21fal 1263 | 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 1264 | 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 1265 | Extend wff definition to include exclusive disjunction ('xor'). |
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Definition | df-xor 1266 |
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 1267 | One way of defining exclusive or. Equivalent to df-xor 1266. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
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Theorem | excxor 1268 | 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 | xorbi2d 1269 | 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 1270 | 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 1271 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
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Theorem | xorbin 1272 | 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 1273 | One direction of pm5.18dc 776, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xornbi 1274 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1279. (Contributed by Jim Kingdon, 10-Mar-2018.) |
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Theorem | xor3dc 1275 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
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Theorem | xorcom 1276 |
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Theorem | pm5.15dc 1277 | 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 1278 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
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Theorem | xornbidc 1279 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
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Theorem | xordc 1280 | 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 1281 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
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Theorem | nbbndc 1282 | 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 1283 |
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 1284 | 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 1285 | 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 1286 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | xordidc 1287 | 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 1288 | 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
| ||
Theorem | truantru 1289 |
A ![]() |
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Theorem | truanfal 1290 |
A ![]() |
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Theorem | falantru 1291 |
A ![]() |
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Theorem | falanfal 1292 |
A ![]() |
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Theorem | truortru 1293 |
A ![]() |
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Theorem | truorfal 1294 |
A ![]() |
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Theorem | falortru 1295 |
A ![]() |
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Theorem | falorfal 1296 |
A ![]() |
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Theorem | truimtru 1297 |
A ![]() |
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Theorem | truimfal 1298 |
A ![]() |
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Theorem | falimtru 1299 |
A ![]() |
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Theorem | falimfal 1300 |
A ![]() |
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