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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3impdi 1201 | Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.) |
Theorem | 3impdir 1202 | Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.) |
Theorem | 3anidm12 1203 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
Theorem | 3anidm13 1204 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
Theorem | 3anidm23 1205 | Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) |
Theorem | 3ori 1206 | Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.) |
Theorem | 3jao 1207 | Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) |
Theorem | 3jaob 1208 | Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) |
Theorem | 3jaoi 1209 | Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.) |
Theorem | 3jaod 1210 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
Theorem | 3jaoian 1211 | Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.) |
Theorem | 3jaodan 1212 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
Theorem | mpjao3dan 1213 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Theorem | 3jaao 1214 | Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | 3ianorr 1215 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
Theorem | syl3an9b 1216 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
Theorem | 3orbi123d 1217 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
Theorem | 3anbi123d 1218 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
Theorem | 3anbi12d 1219 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anbi13d 1220 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anbi23d 1221 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anbi1d 1222 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anbi2d 1223 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anbi3d 1224 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
Theorem | 3anim123d 1225 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
Theorem | 3orim123d 1226 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
Theorem | an6 1227 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
Theorem | 3an6 1228 | Analog of an4 528 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3or6 1229 | Analog of or4 698 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
Theorem | mp3an1 1230 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
Theorem | mp3an2 1231 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
Theorem | mp3an3 1232 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
Theorem | mp3an12 1233 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
Theorem | mp3an13 1234 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
Theorem | mp3an23 1235 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
Theorem | mp3an1i 1236 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
Theorem | mp3anl1 1237 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
Theorem | mp3anl2 1238 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
Theorem | mp3anl3 1239 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
Theorem | mp3anr1 1240 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
Theorem | mp3anr2 1241 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
Theorem | mp3anr3 1242 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
Theorem | mp3an 1243 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
Theorem | mpd3an3 1244 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
Theorem | mpd3an23 1245 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
Theorem | mp3and 1246 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | mp3an12i 1247 | mp3an 1243 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
Theorem | mp3an2i 1248 | mp3an 1243 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
Theorem | mp3an3an 1249 | mp3an 1243 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
Theorem | mp3an2ani 1250 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
Theorem | biimp3a 1251 | Infer implication from a logical equivalence. Similar to biimpa 284. (Contributed by NM, 4-Sep-2005.) |
Theorem | biimp3ar 1252 | Infer implication from a logical equivalence. Similar to biimpar 285. (Contributed by NM, 2-Jan-2009.) |
Theorem | 3anandis 1253 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
Theorem | 3anandirs 1254 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
Theorem | ecased 1255 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
Theorem | ecase23d 1256 | Variation of ecased 1255 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
Even though it isn't ordinarily part of propositional calculus, the universal quantifier is introduced here so that the soundness of definition df-tru 1262 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1352 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1267 may be adopted and this subsection moved down to the start of the subsection with wex 1397 below. However, the use of dftru2 1267 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1257 | Extend wff definition to include the universal quantifier ('for all'). is read " (phi) is true for all ." Typically, in its final application would be replaced with a wff containing a (free) occurrence of the variable , for example . In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of . When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |
Even though it isn't ordinarily part of propositional calculus, the equality predicate is introduced here so that the soundness of definition df-tru 1262 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1411 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1267 may be adopted and this subsection moved down to just above weq 1408 below. However, the use of dftru2 1267 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1258 |
This syntax construction states that a variable , which has been
declared to be a setvar variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder is a class by cab 2042.
Since (when
is distinct from
) we have by
cvjust 2051, we can argue that the syntax " " can be viewed as
an abbreviation for " ". See the
discussion
under the definition of class in [Jech] p.
4 showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1258 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1258 is intrinsically no different from any other class-building syntax such as cab 2042, cun 2942, or c0 3251. For a general discussion of the theory of classes and the role of cv 1258, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1408 of predicate calculus from the wceq 1259 of set theory, so that we don't overload the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) |
Syntax | wceq 1259 |
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 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1408 of predicate calculus in terms of the wceq 1259 of set theory, so that we don't "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 could be the of either weq 1408 or wceq 1259, although mathematically it makes no difference. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2049 for more information on the set theory usage of wceq 1259.) |
Syntax | wtru 1260 | is a wff. |
Theorem | trujust 1261 | Soundness justification theorem for df-tru 1262. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
Definition | df-tru 1262 | Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1263. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1263, and other proofs should depend on tru 1263 (directly or indirectly) instead of this definition, since there are many alternative ways to define . (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.) |
Theorem | tru 1263 | The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
Syntax | wfal 1264 | is a wff. |
Definition | df-fal 1265 | Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1262. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | fal 1266 | The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
Theorem | dftru2 1267 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
Theorem | trud 1268 | Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
Theorem | tbtru 1269 | A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.) |
Theorem | nbfal 1270 | The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.) |
Theorem | bitru 1271 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
Theorem | bifal 1272 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
Theorem | falim 1273 | The truth value implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
Theorem | falimd 1274 | The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | a1tru 1275 | Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
Theorem | truan 1276 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
Theorem | truanOLD 1277 | Obsolete proof of truan 1276 as of 21-Jul-2019. (Contributed by FL, 20-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | dfnot 1278 | Given falsum, we can define the negation of a wff as the statement that a contradiction follows from assuming . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
Theorem | inegd 1279 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | pm2.21fal 1280 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | pclem6 1281 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
Syntax | wxo 1282 | Extend wff definition to include exclusive disjunction ('xor'). |
Definition | df-xor 1283 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with (wa 101), (wo 639), and (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
Theorem | xoranor 1284 | One way of defining exclusive or. Equivalent to df-xor 1283. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
Theorem | excxor 1285 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
Theorem | xoror 1286 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
Theorem | xorbi2d 1287 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi1d 1288 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi12d 1289 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi12i 1290 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Theorem | xorbin 1291 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
Theorem | pm5.18im 1292 | One direction of pm5.18dc 788, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xornbi 1293 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1298. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xor3dc 1294 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID | ||
Theorem | xorcom 1295 | is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
Theorem | pm5.15dc 1296 | 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.) |
DECID DECID | ||
Theorem | xor2dc 1297 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | xornbidc 1298 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
DECID DECID | ||
Theorem | xordc 1299 | 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.) |
DECID DECID | ||
Theorem | xordc1 1300 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
DECID |
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