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	mpjao3dan 1201	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 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.)
|- (ph -> (ps -> ch))   &   |- (th -> (ta -> ch))   &   |- (et -> (ze -> ch))   =>   |- ((ph /\ th /\ et) -> ((ps \/ ta \/ ze) -> ch))
Theorem	3ianorr 1203	Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ((-. ph \/ -. ps \/ -. ch) -> -. (ph /\ ps /\ ch))
Theorem	syl3an9b 1204	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 1205	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 1206	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 1207	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 1208	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 1209	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 1210	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((ps /\ th /\ ta) <-> (ch /\ th /\ ta)))
Theorem	3anbi2d 1211	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ps /\ ta) <-> (th /\ ch /\ ta)))
Theorem	3anbi3d 1212	Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> ((th /\ ta /\ ps) <-> (th /\ ta /\ ch)))
Theorem	3anim123d 1213	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 1214	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 1215	Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
|- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))
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.)
|- (((ph /\ ps) /\ (ch /\ th) /\ (ta /\ et)) <-> ((ph /\ ch /\ ta) /\ (ps /\ th /\ et)))
Theorem	3or6 1217	Analog of or4 687 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
|- (((ph \/ ps) \/ (ch \/ th) \/ (ta \/ et)) <-> ((ph \/ ch \/ ta) \/ (ps \/ th \/ et)))
Theorem	mp3an1 1218	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ph   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mp3an2 1219	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mp3an3 1220	An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 1221	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 1222	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 1223	An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 1224	An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 1225	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 1226	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 1227	An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 1228	An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 1229	An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 1230	An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 1231	An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 1232	An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 1233	An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 1234	Infer implication from a logical equivalence. Similar to biimpa 280. (Contributed by NM, 4-Sep-2005.)
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	biimp3ar 1235	Infer implication from a logical equivalence. Similar to biimpar 281. (Contributed by NM, 2-Jan-2009.)
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ th) -> ch)
Theorem	3anandis 1236	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 1237	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 1238	Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
|- (ph -> -. ch)   &   |- (ph -> (ps \/ ch))   =>   |- (ph -> ps)
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.)
|- (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 1245 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1333 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 1250 may be adopted and this subsection moved down to the start of the subsection with wex 1378 below. However, the use of dftru2 1250 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1240	Extend wff definition to include the universal quantifier ('for all'). A.xph 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.xph
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 1245 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1392 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 1250 may be adopted and this subsection moved down to just above weq 1389 below. However, the use of dftru2 1250 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1241	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 2023. Since (when y is distinct from x) we have x = {y | y e. x } by cvjust 2032, 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 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 class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1389 of predicate calculus from the wceq 1242 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 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 wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1389 of predicate calculus in terms of the wceq 1242 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 1389 or wceq 1242, 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 2030 for more information on the set theory usage of wceq 1242.)
wff A = B
1.2.13.3  Define the true and false constants
Syntax	wtru 1243	T. is a wff.
wff T.
Theorem	trujust 1244	Soundness justification theorem for df-tru 1245. (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 1245	Definition of the truth value "true", or "verum", denoted by T.. This is a tautology, as proved by tru 1246. 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 1246, and other proofs should depend on tru 1246 (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 1246	The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
Syntax	wfal 1247	F. is a wff.
wff F.
Definition	df-fal 1248	Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1245. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	fal 1249	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 1250	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 1251	Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph)   =>   |- ph
Theorem	tbtru 1252	A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- (ph <-> (ph <-> T. ))
Theorem	nbfal 1253	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 1254	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>   |- (ph <-> T. )
Theorem	bifal 1255	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>   |- (ph <-> F. )
Theorem	falim 1256	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 1257	The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ((ph /\ F. ) -> ps)
Theorem	a1tru 1258	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- (ph -> T. )
Theorem	truan 1259	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 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.)
|- (( T. /\ ph) <-> ph)
Theorem	dfnot 1261	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 1262	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ((ph /\ ps) -> F. )   =>   |- (ph -> -. ps)
Theorem	pm2.21fal 1263	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 1264	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 1265	Extend wff definition to include exclusive disjunction ('xor').
wff (ph \/_ ps)
Definition	df-xor 1266	Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with /\ (wa 97), \/ (wo 628), and -> (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
|- ((ph \/_ ps) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
Theorem	xoranor 1267	One way of defining exclusive or. Equivalent to df-xor 1266. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
|- ((ph \/_ ps) <-> ((ph \/ ps) /\ (-. ph \/ -. ps)))
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.)
|- ((ph \/_ ps) <-> ((ph /\ -. ps) \/ (-. ph /\ ps)))
Theorem	xorbi2d 1269	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 1270	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 1271	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	xorbin 1272	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 1273	One direction of pm5.18dc 776, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
|- ((ph <-> ps) -> -. (ph <-> -. ps))
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.)
|- ((ph \/_ ps) -> -. (ph <-> ps))
Theorem	xor3dc 1275	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 1276	\/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
|- ((ph \/_ ps) <-> (ps \/_ ph))
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.)
|- (DECID ph -> (DECID ps -> ((ph <-> ps) \/ (ph <-> -. ps))))
Theorem	xor2dc 1278	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 1279	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 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.)
|- (DECID ph -> (DECID ps -> (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))))
Theorem	xordc1 1281	Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
|- ((ph \/_ ps) -> DECID ph)
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.)
|- (DECID ph -> (DECID ps -> ((-. ph <-> ps) <-> -. (ph <-> ps))))
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.)
|- (DECID ph -> (DECID ps -> (DECID ch -> (((ph <-> ps) <-> ch) <-> (ph <-> (ps <-> ch))))))
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.)
|- (((DECID ph /\ DECID ps) /\ DECID ch) -> ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch))))
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.)
|- (DECID ph -> (DECID ps -> ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))))
Theorem	pm5.24dc 1286	Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
|- (DECID ph -> (DECID ps -> (-. ((ph /\ ps) \/ (-. ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))))
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.)
|- (DECID ph -> (DECID ps -> (DECID ch -> ((ph /\ (ps \/_ ch)) <-> ((ph /\ ps) \/_ (ph /\ ch))))))
Theorem	anxordi 1288	Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
|- ((ph /\ (ps \/_ ch)) <-> ((ph /\ ps) \/_ (ph /\ ch)))
1.2.15  Truth tables: Operations on true and false constants 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 (T.) and false (F.). Although the intuitionistic logic connectives are not as simply defined, T. and F. do play similar roles as in classical logic and most theorems from classical logic continue to hold. Here we show that our definitions and axioms produce equivalent results for T. and F. as we would get from truth tables for /\ (conjunction aka logical 'and') wa 97, \/ (disjunction aka logical inclusive 'or') wo 628, -> (implies) wi 4, -. (not) wn 3, <-> (logical equivalence) df-bi 110, and \/_ (exclusive or) df-xor 1266.
Theorem	truantru 1289	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. /\ T. ) <-> T. )
Theorem	truanfal 1290	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. /\ F. ) <-> F. )
Theorem	falantru 1291	A /\ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
|- (( F. /\ T. ) <-> F. )
Theorem	falanfal 1292	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. /\ F. ) <-> F. )
Theorem	truortru 1293	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. \/ T. ) <-> T. )
Theorem	truorfal 1294	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. \/ F. ) <-> T. )
Theorem	falortru 1295	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. \/ T. ) <-> T. )
Theorem	falorfal 1296	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. \/ F. ) <-> F. )
Theorem	truimtru 1297	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. -> T. ) <-> T. )
Theorem	truimfal 1298	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. -> F. ) <-> F. )
Theorem	falimtru 1299	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. -> T. ) <-> T. )
Theorem	falimfal 1300	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. -> F. ) <-> T. )