P. 13 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1201-1300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3impdi 1201	Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
|- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> th )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	3impdir 1202	Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
|- ( ( ( ph /\ ps ) /\ ( ch /\ ps ) ) -> th )   =>    |- ( ( ph /\ ch /\ ps ) -> th )
Theorem	3anidm12 1203	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
|- ( ( ph /\ ph /\ ps ) -> ch )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	3anidm13 1204	Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
|- ( ( ph /\ ps /\ ph ) -> ch )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	3anidm23 1205	Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
|- ( ( ph /\ ps /\ ps ) -> ch )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	3ori 1206	Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
|- ( ph \/ ps \/ ch )   =>    |- ( ( -. ph /\ -. ps ) -> ch )
Theorem	3jao 1207	Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
|- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) -> ( ( ph \/ ch \/ th ) -> ps ) )
Theorem	3jaob 1208	Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
|- ( ( ( ph \/ ch \/ th ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) )
Theorem	3jaoi 1209	Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
|- ( ph -> ps )   &    |- ( ch -> ps )   &    |- ( th -> ps )   =>    |- ( ( ph \/ ch \/ th ) -> ps )
Theorem	3jaod 1210	Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ch ) )   &    |- ( ph -> ( ta -> ch ) )   =>    |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) )
Theorem	3jaoian 1211	Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( th /\ ps ) -> ch )   &    |- ( ( ta /\ ps ) -> ch )   =>    |- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch )
Theorem	3jaodan 1212	Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   &    |- ( ( ph /\ ta ) -> ch )   =>    |- ( ( ph /\ ( ps \/ th \/ ta ) ) -> ch )
Theorem	mpjao3dan 1213	Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   &    |- ( ( ph /\ ta ) -> ch )   &    |- ( ph -> ( ps \/ th \/ ta ) )   =>    |- ( ph -> ch )
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.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ta -> ch ) )   &    |- ( et -> ( ze -> ch ) )   =>    |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) )
Theorem	3ianorr 1215	Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ( ( -. ph \/ -. ps \/ -. ch ) -> -. ( ph /\ ps /\ ch ) )
Theorem	syl3an9b 1216	Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( th -> ( ch <-> ta ) )   &    |- ( et -> ( ta <-> ze ) )   =>    |- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) )
Theorem	3orbi123d 1217	Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   &    |- ( ph -> ( et <-> ze ) )   =>    |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) )
Theorem	3anbi123d 1218	Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   &    |- ( ph -> ( et <-> ze ) )   =>    |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ ze ) ) )
Theorem	3anbi12d 1219	Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) )
Theorem	3anbi13d 1220	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 1221	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 1222	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps /\ th /\ ta ) <-> ( ch /\ th /\ ta ) ) )
Theorem	3anbi2d 1223	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) )
Theorem	3anbi3d 1224	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) )
Theorem	3anim123d 1225	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 1226	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 1227	Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
|- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) )
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.)
|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) /\ ( ta /\ et ) ) <-> ( ( ph /\ ch /\ ta ) /\ ( ps /\ th /\ et ) ) )
Theorem	3or6 1229	Analog of or4 698 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <-> ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) )
Theorem	mp3an1 1230	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ph   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ps /\ ch ) -> th )
Theorem	mp3an2 1231	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ps   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	mp3an3 1232	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mp3an12 1233	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
|- ph   &    |- ps   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ch -> th )
Theorem	mp3an13 1234	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ph   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ps -> th )
Theorem	mp3an23 1235	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ps   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	mp3an1i 1236	An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
|- ps   &    |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) )   =>    |- ( ph -> ( ( ch /\ th ) -> ta ) )
Theorem	mp3anl1 1237	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ph   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ps /\ ch ) /\ th ) -> ta )
Theorem	mp3anl2 1238	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ps   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ch ) /\ th ) -> ta )
Theorem	mp3anl3 1239	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ch   &    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ps ) /\ th ) -> ta )
Theorem	mp3anr1 1240	An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
|- ps   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ch /\ th ) ) -> ta )
Theorem	mp3anr2 1241	An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
|- ch   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ps /\ th ) ) -> ta )
Theorem	mp3anr3 1242	An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
|- th   &    |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> ta )
Theorem	mp3an 1243	An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
|- ph   &    |- ps   &    |- ch   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- th
Theorem	mpd3an3 1244	An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpd3an23 1245	An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	mp3and 1246	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 1247	mp3an 1243 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 1248	mp3an 1243 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 1249	mp3an 1243 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 1250	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 1251	Infer implication from a logical equivalence. Similar to biimpa 284. (Contributed by NM, 4-Sep-2005.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	biimp3ar 1252	Infer implication from a logical equivalence. Similar to biimpar 285. (Contributed by NM, 2-Jan-2009.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ( ph /\ ps /\ th ) -> ch )
Theorem	3anandis 1253	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 1254	Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
|- ( ( ( ph /\ th ) /\ ( ps /\ th ) /\ ( ch /\ th ) ) -> ta )   =>    |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta )
Theorem	ecased 1255	Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
|- ( ph -> -. ch )   &    |- ( ph -> ( ps \/ ch ) )   =>    |- ( ph -> ps )
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.)
|- ( ph -> -. ch )   &    |- ( ph -> -. th )   &    |- ( ph -> ( ps \/ ch \/ th ) )   =>    |- ( ph -> ps )
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 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'). 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 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 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 2042. Since (when y is distinct from x) we have x = { y | y e. x } by cvjust 2051, 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 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 class x 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.)
class x
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 wff A = B 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 x = y could be the = of either weq 1408 or wceq 1259, 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 2049 for more information on the set theory usage of wceq 1259.)
wff A = B
1.2.13.3  Define the true and false constants
Syntax	wtru 1260	T. is a wff.
wff T.
Theorem	trujust 1261	Soundness justification theorem for df-tru 1262. (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 1262	Definition of the truth value "true", or "verum", denoted by T.. 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 T.. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.)
|- ( T. <-> ( A. x x = x -> A. x x = x ) )
Theorem	tru 1263	The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
Syntax	wfal 1264	F. is a wff.
wff F.
Definition	df-fal 1265	Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1262. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	fal 1266	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 1267	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	trud 1268	Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph )   =>    |- ph
Theorem	tbtru 1269	A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( ph <-> ( ph <-> T. ) )
Theorem	nbfal 1270	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 1271	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>    |- ( ph <-> T. )
Theorem	bifal 1272	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>    |- ( ph <-> F. )
Theorem	falim 1273	The truth value F. 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.)
|- ( F. -> ph )
Theorem	falimd 1274	The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ F. ) -> ps )
Theorem	a1tru 1275	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( ph -> T. )
Theorem	truan 1276	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
|- ( ( T. /\ ph ) <-> ph )
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.)
|- ( ( T. /\ ph ) <-> ph )
Theorem	dfnot 1278	Given falsum, we can define the negation of a wff ph as the statement that a contradiction follows from assuming ph. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
|- ( -. ph <-> ( ph -> F. ) )
Theorem	inegd 1279	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ ps ) -> F. )   =>    |- ( ph -> -. ps )
Theorem	pm2.21fal 1280	If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> ps )   &    |- ( ph -> -. ps )   =>    |- ( ph -> F. )
Theorem	pclem6 1281	Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
|- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps )
1.2.14  Logical 'xor'
Syntax	wxo 1282	Extend wff definition to include exclusive disjunction ('xor').
wff ( ph \/_ ps )
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.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
Theorem	xoranor 1284	One way of defining exclusive or. Equivalent to df-xor 1283. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )
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.)
|- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
Theorem	xoror 1286	XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
|- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
Theorem	xorbi2d 1287	Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th \/_ ps ) <-> ( th \/_ ch ) ) )
Theorem	xorbi1d 1288	Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ th ) ) )
Theorem	xorbi12d 1289	Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
Theorem	xorbi12i 1290	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )
Theorem	xorbin 1291	A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
|- ( ( ph \/_ ps ) -> ( ph <-> -. ps ) )
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.)
|- ( ( ph <-> ps ) -> -. ( ph <-> -. ps ) )
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.)
|- ( ( ph \/_ ps ) -> -. ( ph <-> ps ) )
Theorem	xor3dc 1294	Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )
Theorem	xorcom 1295	\/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
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 ph -> (DECID ps -> ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) ) )
Theorem	xor2dc 1297	Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) )
Theorem	xornbidc 1298	Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )
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 ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )
Theorem	xordc1 1300	Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
|- ( ( ph \/_ ps ) -> DECID ph )