P. 15 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3anbi13d 1401	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) )
Theorem	3anbi23d 1402	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( et /\ ps /\ th ) <-> ( et /\ ch /\ ta ) ) )
Theorem	3anbi1d 1403	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps /\ th /\ ta ) <-> ( ch /\ th /\ ta ) ) )
Theorem	3anbi2d 1404	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) )
Theorem	3anbi3d 1405	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )
Theorem	3anim123d 1406	Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ta ) )   &    |- ( ph -> ( et -> ze ) )   =>    |- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) )
Theorem	3orim123d 1407	Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ta ) )   &    |- ( ph -> ( et -> ze ) )   =>    |- ( ph -> ( ( ps \/ th \/ et ) -> ( ch \/ ta \/ ze ) ) )
Theorem	an6 1408	Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
|- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) )
Theorem	3an6 1409	Analogue of an4 865 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) /\ ( ta /\ et ) ) <-> ( ( ph /\ ch /\ ta ) /\ ( ps /\ th /\ et ) ) )
Theorem	3or6 1410	Analogue of or4 550 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <-> ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) )
Theorem	mp3an1 1411	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ph   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ps /\ ch ) -> th )
Theorem	mp3an2 1412	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ps   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	mp3an3 1413	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mp3an12 1414	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
|- ph   &    |- ps   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ch -> th )
Theorem	mp3an13 1415	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ph   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ps -> th )
Theorem	mp3an23 1416	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ps   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	mp3an1i 1417	An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
|- ps   &    |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) )   =>    |- ( ph -> ( ( ch /\ th ) -> ta ) )
Theorem	mp3anl1 1418	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ph   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ps /\ ch ) /\ th ) -> ta )
Theorem	mp3anl2 1419	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ps   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ch ) /\ th ) -> ta )
Theorem	mp3anl3 1420	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ch   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ps ) /\ th ) -> ta )
Theorem	mp3anr1 1421	An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
|- ps   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ch /\ th ) ) -> ta )
Theorem	mp3anr2 1422	An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
|- ch   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ps /\ th ) ) -> ta )
Theorem	mp3anr3 1423	An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
|- th   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> ta )
Theorem	mp3an 1424	An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
|- ph   &    |- ps   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- th
Theorem	mpd3an3 1425	An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpd3an23 1426	An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	mp3and 1427	A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> th )   &    |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) )   =>    |- ( ph -> ta )
Theorem	mp3an12i 1428	mp3an 1424 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
|- ph   &    |- ps   &    |- ( ch -> th )   &    |- ( ( ph /\ ps /\ th ) -> ta )   =>    |- ( ch -> ta )
Theorem	mp3an2i 1429	mp3an 1424 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
|- ph   &    |- ( ps -> ch )   &    |- ( ps -> th )   &    |- ( ( ph /\ ch /\ th ) -> ta )   =>    |- ( ps -> ta )
Theorem	mp3an3an 1430	mp3an 1424 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
|- ph   &    |- ( ps -> ch )   &    |- ( th -> ta )   &    |- ( ( ph /\ ch /\ ta ) -> et )   =>    |- ( ( ps /\ th ) -> et )
Theorem	mp3an2ani 1431	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
|- ph   &    |- ( ps -> ch )   &    |- ( ( ps /\ th ) -> ta )   &    |- ( ( ph /\ ch /\ ta ) -> et )   =>    |- ( ( ps /\ th ) -> et )
Theorem	biimp3a 1432	Infer implication from a logical equivalence. Similar to biimpa 501. (Contributed by NM, 4-Sep-2005.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	biimp3ar 1433	Infer implication from a logical equivalence. Similar to biimpar 502. (Contributed by NM, 2-Jan-2009.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ( ph /\ ps /\ th ) -> ch )
Theorem	3anandis 1434	Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
|- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )
Theorem	3anandirs 1435	Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)
|- ( ( ( ph /\ th ) /\ ( ps /\ th ) /\ ( ch /\ th ) ) -> ta )   =>    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )
Theorem	ecase23d 1436	Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
|- ( ph -> -. ch )   &    |- ( ph -> -. th )   &    |- ( ph -> ( ps \/ ch \/ th ) )   =>    |- ( ph -> ps )
Theorem	3ecase 1437	Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
|- ( -. ph -> th )   &    |- ( -. ps -> th )   &    |- ( -. ch -> th )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- th
Theorem	3bior1fd 1438	A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 420. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
|- ( ph -> -. th )   =>    |- ( ph -> ( ( ch \/ ps ) <-> ( th \/ ch \/ ps ) ) )
Theorem	3bior1fand 1439	A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
|- ( ph -> -. th )   =>    |- ( ph -> ( ( ch \/ ps ) <-> ( ( th /\ ta ) \/ ch \/ ps ) ) )
Theorem	3bior2fd 1440	A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 420. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
|- ( ph -> -. th )   &    |- ( ph -> -. ch )   =>    |- ( ph -> ( ps <-> ( th \/ ch \/ ps ) ) )
Theorem	3biant1d 1441	A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 528. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
|- ( ph -> th )   =>    |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ch /\ ps ) ) )
Theorem	intn3an1d 1442	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. ( ps /\ ch /\ th ) )
Theorem	intn3an2d 1443	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. ( ch /\ ps /\ th ) )
Theorem	intn3an3d 1444	Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. ( ch /\ th /\ ps ) )
Theorem	an3andi 1445	Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
|- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) )
Theorem	an33rean 1446	Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.)
|- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\ rh ) ) <-> ( ( ph /\ ta /\ rh ) /\ ( ( ps /\ th ) /\ ( et /\ si ) /\ ( ch /\ ze ) ) ) )
1.2.11  Logical 'nand' (Sheffer stroke)
Syntax	wnan 1447	Extend wff definition to include alternative denial ('nand').
wff ( ph -/\ ps )
Definition	df-nan 1448	Define incompatibility, or alternative denial ('not-and' or 'nand'). This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar. In the second edition of Principia Mathematica (1927), Russell and Whitehead used the Sheffer stroke and suggested it as a replacement for the "or" and "not" operations of the first edition. However, in practice, "or" and "not" are more widely used. After we define the constant true T. (df-tru 1486) and the constant false F. (df-fal 1489), we will be able to prove these truth table values: ( ( T. -/\ T. ) <-> F. ) (trunantru 1524), ( ( T. -/\ F. ) <-> T. ) (trunanfal 1525), ( ( F. -/\ T. ) <-> T. ) (falnantru 1526), and ( ( F. -/\ F. ) <-> T. ) (falnanfal 1527). Contrast with /\ (df-an 386), \/ (df-or 385), -> (wi 4), and \/_ (df-xor 1465) . (Contributed by Jeff Hoffman, 19-Nov-2007.)
|- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
Theorem	nanan 1449	Write 'and' in terms of 'nand'. (Contributed by Mario Carneiro, 9-May-2015.)
|- ( ( ph /\ ps ) <-> -. ( ph -/\ ps ) )
Theorem	nancom 1450	The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
|- ( ( ph -/\ ps ) <-> ( ps -/\ ph ) )
Theorem	nannan 1451	Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
|- ( ( ph -/\ ( ch -/\ ps ) ) <-> ( ph -> ( ch /\ ps ) ) )
Theorem	nanim 1452	Show equivalence between implication and the Nicod version. To derive nic-dfim 1594, apply nanbi 1454. (Contributed by Jeff Hoffman, 19-Nov-2007.)
|- ( ( ph -> ps ) <-> ( ph -/\ ( ps -/\ ps ) ) )
Theorem	nannot 1453	Show equivalence between negation and the Nicod version. To derive nic-dfneg 1595, apply nanbi 1454. (Contributed by Jeff Hoffman, 19-Nov-2007.)
|- ( -. ps <-> ( ps -/\ ps ) )
Theorem	nanbi 1454	Show equivalence between the biconditional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
|- ( ( ph <-> ps ) <-> ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) )
Theorem	nanbi1 1455	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )
Theorem	nanbi2 1456	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) )
Theorem	nanbi12 1457	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) )
Theorem	nanbi1i 1458	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   =>    |- ( ( ph -/\ ch ) <-> ( ps -/\ ch ) )
Theorem	nanbi2i 1459	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   =>    |- ( ( ch -/\ ph ) <-> ( ch -/\ ps ) )
Theorem	nanbi12i 1460	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph -/\ ch ) <-> ( ps -/\ th ) )
Theorem	nanbi1d 1461	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ th ) ) )
Theorem	nanbi2d 1462	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th -/\ ps ) <-> ( th -/\ ch ) ) )
Theorem	nanbi12d 1463	Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) )
1.2.12  Logical 'xor'
Syntax	wxo 1464	Extend wff definition to include exclusive disjunction ('xor').
wff ( ph \/_ ps )
Definition	df-xor 1465	Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true T. (df-tru 1486) and the constant false F. (df-fal 1489), we will be able to prove these truth table values: ( ( T. \/_ T. ) <-> F. ) (truxortru 1528), ( ( T. \/_ F. ) <-> T. ) (truxorfal 1529), ( ( F. \/_ T. ) <-> T. ) (falxortru 1530), and ( ( F. \/_ F. ) <-> F. ) (falxorfal 1531). Contrast with /\ (df-an 386), \/ (df-or 385), -> (wi 4), and -/\ (df-nan 1448) . (Contributed by FL, 22-Nov-2010.)
|- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
Theorem	xnor 1466	Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorcom 1467	The connector \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
Theorem	xorass 1468	The connector \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)
|- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )
Theorem	excxor 1469	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
|- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
Theorem	xor2 1470	Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
Theorem	xoror 1471	XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
|- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
Theorem	xornan 1472	XOR implies NAND. (Contributed by BJ, 19-Apr-2019.)
|- ( ( ph \/_ ps ) -> -. ( ph /\ ps ) )
Theorem	xornan2 1473	XOR implies NAND (written with the -/\ connector). (Contributed by BJ, 19-Apr-2019.)
|- ( ( ph \/_ ps ) -> ( ph -/\ ps ) )
Theorem	xorneg2 1474	The connector \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
|- ( ( ph \/_ -. ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorneg1 1475	The connector \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
|- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorneg 1476	The connector \/_ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( -. ph \/_ -. ps ) <-> ( ph \/_ ps ) )
Theorem	xorbi12i 1477	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )
Theorem	xorbi12d 1478	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
Theorem	anxordi 1479	Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 937 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)
|- ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) )
Theorem	xorexmid 1480	Exclusive-or variant of the law of the excluded middle (exmid 431). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019.)
|- ( ph \/_ -. ph )
1.2.13  True and false constants
1.2.13.1  Universal quantifier for use by df-tru Even though it isn't ordinarily part of propositional calculus, the universal quantifier A. is introduced here so that the soundness of definition df-tru 1486 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in definition df-ex 1705 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 1491 may be adopted and this subsection moved down to the start of the subsection with wex 1704 below. However, the use of dftru2 1491 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1481	Extend wff definition to include the universal quantifier ('for all'). A. x ph is read " ph (phi) is true for all x." Typically, in its final application ph would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. 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 x. 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.
wff A. x ph
1.2.13.2  Equality predicate for use by df-tru Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1486 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in theorem equs3 1875 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 1491 may be adopted and this subsection moved down to just above weq 1874 below. However, the use of dftru2 1491 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1482	This syntax construction states that a variable x, 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 { y | y e. x } is a class by cab 2608. Since (when y is distinct from x) we have x = { y | y e. x } by cvjust 2617, we can argue that the syntax " class x " can be viewed as an abbreviation for " class { y | y e. x }". 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 1482 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1482 is intrinsically no different from any other class-building syntax such as cab 2608, cun 3572, or c0 3915. For a general discussion of the theory of classes and the role of cv 1482, see mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1874 of predicate calculus from the wceq 1483 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.)
class x
Syntax	wceq 1483	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 us to express i.e. "prove" the weq 1874 of predicate calculus in terms of the wceq 1483 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 x = y could be the = of either weq 1874 or wceq 1483, 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 2615 for more information on the set theory usage of wceq 1483.)
wff A = B
1.2.13.3  Define the true and false constants
Syntax	wtru 1484	The constant T. is a wff.
wff T.
Theorem	trujust 1485	Soundness justification theorem for df-tru 1486. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
|- ( ( A. x x = x -> A. x x = x ) <-> ( A. y y = y -> A. y y = y ) )
Definition	df-tru 1486	Definition of the truth value "true", or "verum", denoted by T.. This is a tautology, as proved by tru 1487. In this definition, an instance of id 22 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 22 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 1487, and other proofs should depend on tru 1487 (directly or indirectly) instead of this definition, since there are many alternate ways to define T.. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) Use tru 1487 instead. (New usage is discouraged.)
|- ( T. <-> ( A. x x = x -> A. x x = x ) )
Theorem	tru 1487	The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
Syntax	wfal 1488	The constant F. is a wff.
wff F.
Definition	df-fal 1489	Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1486. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	fal 1490	The truth value F. is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
|- -. F.
Theorem	dftru2 1491	An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
|- ( T. <-> ( ph -> ph ) )
Theorem	trut 1492	A proposition is equivalent to it being implied by T.. Closed form of trud 1493. Dual of dfnot 1502. It is to tbtru 1494 what a1bi 352 is to tbt 359. (Contributed by BJ, 26-Oct-2019.)
|- ( ph <-> ( T. -> ph ) )
Theorem	trud 1493	Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph )   =>    |- ph
Theorem	tbtru 1494	A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( ph <-> ( ph <-> T. ) )
Theorem	nbfal 1495	The negation of a proposition is equivalent to itself being equivalent to F.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( -. ph <-> ( ph <-> F. ) )
Theorem	bitru 1496	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>    |- ( ph <-> T. )
Theorem	bifal 1497	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>    |- ( ph <-> F. )
Theorem	falim 1498	The truth value F. implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( F. -> ph )
Theorem	falimd 1499	The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ F. ) -> ps )
Theorem	a1tru 1500	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( ph -> T. )