P. 15 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-fal 1401	Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1398. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	fal 1402	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 1403	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	mptru 1404	Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph )   =>    |- ph
Theorem	tbtru 1405	A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( ph <-> ( ph <-> T. ) )
Theorem	nbfal 1406	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 1407	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>    |- ( ph <-> T. )
Theorem	bifal 1408	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>    |- ( ph <-> F. )
Theorem	falim 1409	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 1410	The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ F. ) -> ps )
Theorem	trud 1411	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( ph -> T. )
Theorem	truan 1412	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	dfnot 1413	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 1414	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ ps ) -> F. )   =>    |- ( ph -> -. ps )
Theorem	pm2.21fal 1415	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 1416	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 1417	Extend wff definition to include exclusive disjunction ('xor').
wff ( ph \/_ ps )
Definition	df-xor 1418	Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with /\ (wa 104), \/ (wo 713), and -> (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
Theorem	xoranor 1419	One way of defining exclusive or. Equivalent to df-xor 1418. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )
Theorem	excxor 1420	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 1421	XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
|- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
Theorem	xorbi2d 1422	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 1423	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 1424	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 1425	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )
Theorem	xorbin 1426	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 1427	One direction of pm5.18dc 888, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
|- ( ( ph <-> ps ) -> -. ( ph <-> -. ps ) )
Theorem	xornbi 1428	A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1433. (Contributed by Jim Kingdon, 10-Mar-2018.)
|- ( ( ph \/_ ps ) -> -. ( ph <-> ps ) )
Theorem	xor3dc 1429	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 1430	\/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
Theorem	pm5.15dc 1431	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 1432	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 1433	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 1434	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 1435	Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
|- ( ( ph \/_ ps ) -> DECID ph )
Theorem	nbbndc 1436	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 1437	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 1438	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 1439	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 1440	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 1441	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 1442	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 104, \/ (disjunction aka logical inclusive 'or') wo 713, -> (implies) wi 4, -. (not) wn 3, <-> (logical equivalence) df-bi 117, and \/_ (exclusive or) df-xor 1418.
Theorem	truantru 1443	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ T. ) <-> T. )
Theorem	truanfal 1444	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ F. ) <-> F. )
Theorem	falantru 1445	A /\ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
|- ( ( F. /\ T. ) <-> F. )
Theorem	falanfal 1446	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. /\ F. ) <-> F. )
Theorem	truortru 1447	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. \/ T. ) <-> T. )
Theorem	truorfal 1448	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. \/ F. ) <-> T. )
Theorem	falortru 1449	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. \/ T. ) <-> T. )
Theorem	falorfal 1450	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. \/ F. ) <-> F. )
Theorem	truimtru 1451	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. -> T. ) <-> T. )
Theorem	truimfal 1452	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. -> F. ) <-> F. )
Theorem	falimtru 1453	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> T. ) <-> T. )
Theorem	falimfal 1454	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> F. ) <-> T. )
Theorem	nottru 1455	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( -. T. <-> F. )
Theorem	notfal 1456	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( -. F. <-> T. )
Theorem	trubitru 1457	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. <-> T. ) <-> T. )
Theorem	trubifal 1458	A <-> identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
|- ( ( T. <-> F. ) <-> F. )
Theorem	falbitru 1459	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> T. ) <-> F. )
Theorem	falbifal 1460	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> F. ) <-> T. )
Theorem	truxortru 1461	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( T. \/_ T. ) <-> F. )
Theorem	truxorfal 1462	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( T. \/_ F. ) <-> T. )
Theorem	falxortru 1463	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( F. \/_ T. ) <-> T. )
Theorem	falxorfal 1464	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( F. \/_ F. ) <-> F. )
1.2.16  Stoic logic indemonstrables (Chrysippus of Soli) The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara and related theorems. A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 666, modus ponendo tollens I mptnan 1465, modus ponendo tollens II mptxor 1466, and modus tollendo ponens (exclusive-or version) mtpxor 1468. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtpxor 1468 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1467. This set of indemonstrables is not the entire system of Stoic logic.
Theorem	mptnan 1465	Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1466) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
|- ph   &    |- -. ( ph /\ ps )   =>    |- -. ps
Theorem	mptxor 1466	Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or \/_. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ph   &    |- ( ph \/_ ps )   =>    |- -. ps
Theorem	mtpor 1467	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1468, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if ph is not true, and ph or ps (or both) are true, then ps must be true". An alternate phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth". -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
|- -. ph   &    |- ( ph \/ ps )   =>    |- ps
Theorem	mtpxor 1468	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1467, one of the five "indemonstrables" in Stoic logic. The rule says, "if ph is not true, and either ph or ps (exclusively) are true, then ps must be true". Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1467. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1466, that is, it is exclusive-or df-xor 1418), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1466), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
|- -. ph   &    |- ( ph \/_ ps )   =>    |- ps
Theorem	stoic1a 1469	Stoic logic Thema 1 (part a). The first thema of the four Stoic logic themata, in its basic form, was: "When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/ We will represent thema 1 as two very similar rules stoic1a 1469 and stoic1b 1470 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ph /\ -. th ) -> -. ps )
Theorem	stoic1b 1470	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1469. (Contributed by David A. Wheeler, 16-Feb-2019.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ps /\ -. th ) -> -. ph )
Theorem	stoic2a 1471	Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as ph, ps |- ch; in Metamath we will represent that construct as ph /\ ps -> ch. This version a is without the phrase "or both"; see stoic2b 1472 for the version with the phrase "or both". We already have this rule as syldan 282, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	stoic2b 1472	Stoic logic Thema 2 version b. See stoic2a 1471. Version b is with the phrase "or both". We already have this rule as mpd3an3 1372, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ps /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	stoic3 1473	Stoic logic Thema 3. Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ch /\ th ) -> ta )   =>    |- ( ( ph /\ ps /\ th ) -> ta )
Theorem	stoic4a 1474	Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)." We use th to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1475 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ch /\ ph /\ th ) -> ta )   =>    |- ( ( ph /\ ps /\ th ) -> ta )
Theorem	stoic4b 1475	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1474 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ( ch /\ ph /\ ps ) /\ th ) -> ta )   =>    |- ( ( ph /\ ps /\ th ) -> ta )
1.2.17  Logical implication (continued)
Theorem	syl6an 1476	A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
|- ( ph -> ps )   &    |- ( ph -> ( ch -> th ) )   &    |- ( ( ps /\ th ) -> ta )   =>    |- ( ph -> ( ch -> ta ) )
Theorem	syl10 1477	A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ps -> ( th -> ta ) ) )   &    |- ( ch -> ( ta -> et ) )   =>    |- ( ph -> ( ps -> ( th -> et ) ) )
Theorem	a1ddd 1478	Triple deduction introducing an antecedent to a wff. Deduction associated with a1dd 48. Double deduction associated with a1d 22. Triple deduction associated with ax-1 6 and a1i 9. (Contributed by Jeff Hankins, 4-Aug-2009.)
|- ( ph -> ( ps -> ( ch -> ta ) ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exbir 1479	Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
Theorem	3impexp 1480	impexp 263 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
Theorem	3impexpbicom 1481	3impexp 1480 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
Theorem	3impexpbicomi 1482	Deduction form of 3impexpbicom 1481. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
Theorem	ancomsimp 1483	Closed form of ancoms 268. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
Theorem	expcomd 1484	Deduction form of expcom 116. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> ( ps -> th ) ) )
Theorem	expdcom 1485	Commuted form of expd 258. (Contributed by Alan Sare, 18-Mar-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ps -> ( ch -> ( ph -> th ) ) )
Theorem	simplbi2comg 1486	Implication form of simplbi2com 1487. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
Theorem	simplbi2com 1487	A deduction eliminating a conjunct, similar to simplbi2 385. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ps -> ph ) )
Theorem	syl6ci 1488	A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> th )   &    |- ( ch -> ( th -> ta ) )   =>    |- ( ph -> ( ps -> ta ) )
Theorem	mpisyl 1489	A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
|- ( ph -> ps )   &    |- ch   &    |- ( ps -> ( ch -> th ) )   =>    |- ( ph -> th )
Theorem	dcfromnotnotr 1490	The decidability of a proposition ps follows from a suitable instance of double negation elimination (DNE). Therefore, if we were to introduce DNE as a general principle (without the decidability condition in notnotrdc 848), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since DNE itself is classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ps \/ -. ps ) )   &    |- ( -. -. ph -> ph )   =>    |- DECID ps
Theorem	dcfromcon 1491	The decidability of a proposition ch follows from a suitable instance of the principle of contraposition. Therefore, if we were to introduce contraposition as a general principle (without the decidability condition in condc 858), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since the principle of contraposition is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ch \/ -. ch ) )   &    |- ( ps <-> T. )   &    |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) )   =>    |- DECID ch
Theorem	dcfrompeirce 1492	The decidability of a proposition ch follows from a suitable instance of Peirce's law. Therefore, if we were to introduce Peirce's law as a general principle (without the decidability condition in peircedc 919), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since Perice's law is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ch \/ -. ch ) )   &    |- ( ps <-> F. )   &    |- ( ( ( ph -> ps ) -> ph ) -> ph )   =>    |- DECID ch
1.3  Predicate calculus mostly without distinct variables
1.3.1  Universal quantifier (continued) The universal quantifier was introduced above in wal 1393 for use by df-tru 1398. See the comments in that section. In this section, we continue with the first "real" use of it.
Axiom	ax-5 1493	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Axiom	ax-7 1494	Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph -> A. y A. x ph )
Axiom	ax-gen 1495	Rule of Generalization. The postulated inference rule of predicate calculus. See, e.g., Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved x = x, we can conclude A. x x = x or even A. y x = x. Theorem spi 1582 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.)
|- ph   =>    |- A. x ph
Theorem	gen2 1496	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
|- ph   =>    |- A. x A. y ph
Theorem	mpg 1497	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
|- ( A. x ph -> ps )   &    |- ph   =>    |- ps
Theorem	mpgbi 1498	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
|- ( A. x ph <-> ps )   &    |- ph   =>    |- ps
Theorem	mpgbir 1499	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
|- ( ph <-> A. x ps )   &    |- ps   =>    |- ph
Theorem	a7s 1500	Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )