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
Theorem	xorbi12d 1401	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 1402	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )
Theorem	xorbin 1403	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 1404	One direction of pm5.18dc 884, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
|- ( ( ph <-> ps ) -> -. ( ph <-> -. ps ) )
Theorem	xornbi 1405	A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1410. (Contributed by Jim Kingdon, 10-Mar-2018.)
|- ( ( ph \/_ ps ) -> -. ( ph <-> ps ) )
Theorem	xor3dc 1406	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 1407	\/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
Theorem	pm5.15dc 1408	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 1409	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 1410	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 1411	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 1412	Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
|- ( ( ph \/_ ps ) -> DECID ph )
Theorem	nbbndc 1413	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 1414	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 1415	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 1416	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 1417	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 1418	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 1419	Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
|- ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) )
1.2.14  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 709, -> (implies) wi 4, -. (not) wn 3, <-> (logical equivalence) df-bi 117, and \/_ (exclusive or) df-xor 1395.
Theorem	truantru 1420	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ T. ) <-> T. )
Theorem	truanfal 1421	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ F. ) <-> F. )
Theorem	falantru 1422	A /\ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
|- ( ( F. /\ T. ) <-> F. )
Theorem	falanfal 1423	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. /\ F. ) <-> F. )
Theorem	truortru 1424	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. \/ T. ) <-> T. )
Theorem	truorfal 1425	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. \/ F. ) <-> T. )
Theorem	falortru 1426	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. \/ T. ) <-> T. )
Theorem	falorfal 1427	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. \/ F. ) <-> F. )
Theorem	truimtru 1428	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. -> T. ) <-> T. )
Theorem	truimfal 1429	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. -> F. ) <-> F. )
Theorem	falimtru 1430	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> T. ) <-> T. )
Theorem	falimfal 1431	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> F. ) <-> T. )
Theorem	nottru 1432	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( -. T. <-> F. )
Theorem	notfal 1433	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( -. F. <-> T. )
Theorem	trubitru 1434	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. <-> T. ) <-> T. )
Theorem	trubifal 1435	A <-> identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
|- ( ( T. <-> F. ) <-> F. )
Theorem	falbitru 1436	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> T. ) <-> F. )
Theorem	falbifal 1437	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> F. ) <-> T. )
Theorem	truxortru 1438	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( T. \/_ T. ) <-> F. )
Theorem	truxorfal 1439	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( T. \/_ F. ) <-> T. )
Theorem	falxortru 1440	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( F. \/_ T. ) <-> T. )
Theorem	falxorfal 1441	A \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
|- ( ( F. \/_ F. ) <-> F. )
1.2.15  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 663, modus ponendo tollens I mptnan 1442, modus ponendo tollens II mptxor 1443, and modus tollendo ponens (exclusive-or version) mtpxor 1445. 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 1445 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 1444. This set of indemonstrables is not the entire system of Stoic logic.
Theorem	mptnan 1442	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 1443) 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 1443	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 1444	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1445, 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 1445	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1444, 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 1444. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1443, that is, it is exclusive-or df-xor 1395), 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 1443), 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 1446	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 1446 and stoic1b 1447 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 1447	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1446. (Contributed by David A. Wheeler, 16-Feb-2019.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ps /\ -. th ) -> -. ph )
Theorem	stoic2a 1448	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 1449 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 1449	Stoic logic Thema 2 version b. See stoic2a 1448. Version b is with the phrase "or both". We already have this rule as mpd3an3 1350, 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 1450	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 1451	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 1452 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 1452	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1451 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ( ch /\ ph /\ ps ) /\ th ) -> ta )   =>    |- ( ( ph /\ ps /\ th ) -> ta )
1.2.16  Logical implication (continued)
Theorem	syl6an 1453	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 1454	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	exbir 1455	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 1456	impexp 263 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
Theorem	3impexpbicom 1457	3impexp 1456 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 1458	Deduction form of 3impexpbicom 1457. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
Theorem	ancomsimp 1459	Closed form of ancoms 268. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
Theorem	expcomd 1460	Deduction form of expcom 116. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> ( ps -> th ) ) )
Theorem	expdcom 1461	Commuted form of expd 258. (Contributed by Alan Sare, 18-Mar-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ps -> ( ch -> ( ph -> th ) ) )
Theorem	simplbi2comg 1462	Implication form of simplbi2com 1463. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
Theorem	simplbi2com 1463	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 1464	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 1465	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 1466	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 844), 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 1467	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 854), 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 1468	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 915), 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 1370 for use by df-tru 1375. See the comments in that section. In this section, we continue with the first "real" use of it.
Axiom	ax-5 1469	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 1470	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 1471	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 1558 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 1472	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
|- ph   =>    |- A. x A. y ph
Theorem	mpg 1473	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
|- ( A. x ph -> ps )   &    |- ph   =>    |- ps
Theorem	mpgbi 1474	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 1475	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 1476	Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	alimi 1477	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	2alimi 1478	Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> ps )   =>    |- ( A. x A. y ph -> A. x A. y ps )
Theorem	alim 1479	Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	al2imi 1480	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x ph -> ( A. x ps -> A. x ch ) )
Theorem	alanimi 1481	Variant of al2imi 1480 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x ph /\ A. x ps ) -> A. x ch )
Syntax	wnf 1482	Extend wff definition to include the not-free predicate.
wff F/ x ph
Definition	df-nf 1483	Define the not-free predicate for wffs. This is read " x is not free in ph". Not-free means that the value of x cannot affect the value of ph, e.g., any occurrence of x in ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1799). An example of where this is used is stdpc5 1606. See nf2 1690 for an alternate definition which does not involve nested quantifiers on the same variable. Nonfreeness is a commonly used condition, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the notion of nonfreeness within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for "not free" (which considers syntactic freedom). For example, x is effectively not free in the expression x = x (even though x is syntactically free in it, so would be considered "free" in the usual textbook definition) because the value of x in the formula x = x does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 1724. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
Theorem	nfi 1484	Deduce that x is not free in ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> A. x ph )   =>    |- F/ x ph
Theorem	hbth 1485	No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form |- ( ph -> A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in ph". (Contributed by NM, 5-Aug-1993.)
|- ph   =>    |- ( ph -> A. x ph )
Theorem	nfth 1486	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- F/ x ph
Theorem	nfnth 1487	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
|- -. ph   =>    |- F/ x ph
Theorem	nftru 1488	The true constant has no free variables. (This can also be proven in one step with nfv 1550, but this proof does not use ax-17 1548.) (Contributed by Mario Carneiro, 6-Oct-2016.)
|- F/ x T.
Theorem	alimdh 1489	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	albi 1490	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) )
Theorem	alrimih 1491	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	albii 1492	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
|- ( ph <-> ps )   =>    |- ( A. x ph <-> A. x ps )
Theorem	2albii 1493	Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
|- ( ph <-> ps )   =>    |- ( A. x A. y ph <-> A. x A. y ps )
Theorem	hbxfrbi 1494	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ps )   &    |- ( ps -> A. x ps )   =>    |- ( ph -> A. x ph )
Theorem	nfbii 1495	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   =>    |- ( F/ x ph <-> F/ x ps )
Theorem	nfxfr 1496	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   &    |- F/ x ps   =>    |- F/ x ph
Theorem	nfxfrd 1497	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch -> F/ x ps )   =>    |- ( ch -> F/ x ph )
Theorem	alcoms 1498	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	hbal 1499	If x is not free in ph, it is not free in A. y ph. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( A. y ph -> A. x A. y ph )
Theorem	alcom 1500	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph <-> A. y A. x ph )