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
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 651, modus ponendo tollens I mptnan 1401, modus ponendo tollens II mptxor 1402, and modus tollendo ponens (exclusive-or version) mtpxor 1404. 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 1404 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 1403. This set of indemonstrables is not the entire system of Stoic logic.
Theorem	mptnan 1401	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 1402) 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 1402	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 1403	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1404, 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 1404	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1403, 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 1403. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1402, that is, it is exclusive-or df-xor 1354), 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 1402), 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	stoic2a 1405	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 1406 for the version with the phrase "or both". We already have this rule as syldan 280, 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 1406	Stoic logic Thema 2 version b. See stoic2a 1405. Version b is with the phrase "or both". We already have this rule as mpd3an3 1316, 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 1407	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 1408	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 1409 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 1409	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1408 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 1410	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 1411	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 1412	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 1413	impexp 261 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
Theorem	3impexpbicom 1414	3impexp 1413 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 1415	Deduction form of 3impexpbicom 1414. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
Theorem	ancomsimp 1416	Closed form of ancoms 266. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
Theorem	expcomd 1417	Deduction form of expcom 115. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> ( ps -> th ) ) )
Theorem	expdcom 1418	Commuted form of expd 256. (Contributed by Alan Sare, 18-Mar-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ps -> ( ch -> ( ph -> th ) ) )
Theorem	simplbi2comg 1419	Implication form of simplbi2com 1420. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
Theorem	simplbi2com 1420	A deduction eliminating a conjunct, similar to simplbi2 382. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ps -> ph ) )
Theorem	syl6ci 1421	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 1422	A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
|- ( ph -> ps )   &    |- ch   &    |- ( ps -> ( ch -> th ) )   =>    |- ( ph -> th )
1.3  Predicate calculus mostly without distinct variables
1.3.1  Universal quantifier (continued) The universal quantifier was introduced above in wal 1329 for use by df-tru 1334. See the comments in that section. In this section, we continue with the first "real" use of it.
Axiom	ax-5 1423	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 1424	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 1425	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 1516 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 1426	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
|- ph   =>    |- A. x A. y ph
Theorem	mpg 1427	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
|- ( A. x ph -> ps )   &    |- ph   =>    |- ps
Theorem	mpgbi 1428	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 1429	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 1430	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 1431	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	2alimi 1432	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 1433	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 1434	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 1435	Variant of al2imi 1434 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x ph /\ A. x ps ) -> A. x ch )
Syntax	wnf 1436	Extend wff definition to include the not-free predicate.
wff F/ x ph
Definition	df-nf 1437	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 1750). An example of where this is used is stdpc5 1563. See nf2 1646 for an alternate definition which does not involve nested quantifiers on the same variable. Not-free is a commonly used constraint, 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 not-free notion 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 only considers syntactic freedom). For example, x is effectively not free in the bare expression x = x, even though x would be considered free in the usual textbook definition, because the value of x in the expression x = x cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
Theorem	nfi 1438	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 1439	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 1440	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- F/ x ph
Theorem	nfnth 1441	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
|- -. ph   =>    |- F/ x ph
Theorem	nftru 1442	The true constant has no free variables. (This can also be proven in one step with nfv 1508, but this proof does not use ax-17 1506.) (Contributed by Mario Carneiro, 6-Oct-2016.)
|- F/ x T.
Theorem	alimdh 1443	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 1444	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 1445	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 1446	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 1447	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 1448	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 1449	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   =>    |- ( F/ x ph <-> F/ x ps )
Theorem	nfxfr 1450	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 1451	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 1452	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 1453	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 1454	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph <-> A. y A. x ph )
Theorem	alrimdh 1455	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	albidh 1456	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	19.26 1457	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
Theorem	19.26-2 1458	Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
|- ( A. x A. y ( ph /\ ps ) <-> ( A. x A. y ph /\ A. x A. y ps ) )
Theorem	19.26-3an 1459	Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
|- ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )
Theorem	19.33 1460	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	alrot3 1461	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y A. z ph <-> A. y A. z A. x ph )
Theorem	alrot4 1462	Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)
|- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph )
Theorem	albiim 1463	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
|- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
Theorem	2albiim 1464	Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
|- ( A. x A. y ( ph <-> ps ) <-> ( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) )
Theorem	hband 1465	Deduction form of bound-variable hypothesis builder hban 1526. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> ( ps -> A. x ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) )
Theorem	hb3and 1466	Deduction form of bound-variable hypothesis builder hb3an 1529. (Contributed by NM, 17-Feb-2013.)
|- ( ph -> ( ps -> A. x ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( th -> A. x th ) )   =>    |- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) )
Theorem	hbald 1467	Deduction form of bound-variable hypothesis builder hbal 1453. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( A. y ps -> A. x A. y ps ) )
Syntax	wex 1468	Extend wff definition to include the existential quantifier ("there exists").
wff E. x ph
Axiom	ax-ie1 1469	x is bound in E. x ph. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( E. x ph -> A. x E. x ph )
Axiom	ax-ie2 1470	Define existential quantification. E. x ph means "there exists at least one set x such that ph is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	hbe1 1471	x is not free in E. x ph. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ph -> A. x E. x ph )
Theorem	nfe1 1472	x is not free in E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x E. x ph
Theorem	19.23ht 1473	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23h 1474	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
|- ( ps -> A. x ps )   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	alnex 1475	Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if ph can be refuted for all x, then it is not possible to find an x for which ph holds" (and likewise for the converse). Comparing this with dfexdc 1477 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.)
|- ( A. x -. ph <-> -. E. x ph )
Theorem	nex 1476	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
|- -. ph   =>    |- -. E. x ph
Theorem	dfexdc 1477	Defining E. x ph given decidability. It is common in classical logic to define E. x ph as -. A. x -. ph but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1478. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID E. x ph -> ( E. x ph <-> -. A. x -. ph ) )
Theorem	exalim 1478	One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1477. (Contributed by Jim Kingdon, 29-Jul-2018.)
|- ( E. x ph -> -. A. x -. ph )
1.3.2  Equality predicate (continued) The equality predicate was introduced above in wceq 1331 for use by df-tru 1334. See the comments in that section. In this section, we continue with the first "real" use of it.
Theorem	weq 1479	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1479 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1331. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1479 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1331. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)
wff x = y
Syntax	wcel 1480	Extend wff definition to include the membership connective between classes. (The purpose of introducing wff A e. B here is to allow us to express i.e. "prove" the wel 1481 of predicate calculus in terms of the wceq 1331 of set theory, so that we don't "overload" the e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.)
wff A e. B
Theorem	wel 1481	Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " x is an element of y," " x is a member of y," " x belongs to y," or " y contains x." Note: The phrase " y includes x " means " x is a subset of y;" to use it also for x e. y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactical construction introduces a binary non-logical predicate symbol e. (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 1481 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1480. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1481 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1480. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)
wff x e. y
Axiom	ax-8 1482	Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1685). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. Axioms ax-8 1482 through ax-16 1786 are the axioms having to do with equality, substitution, and logical properties of our binary predicate e. (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1786 and ax-17 1506 are still valid even when x, y, and z are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1786 and ax-17 1506 only. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z -> y = z ) )
Axiom	ax-10 1483	Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). The original version of this axiom was ax-10o 1694 ("o" for "old") and was replaced with this shorter ax-10 1483 in May 2008. The old axiom is proved from this one as theorem ax10o 1693. Conversely, this axiom is proved from ax-10o 1694 as theorem ax10 1695. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> A. y y = x )
Axiom	ax-11 1484	Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A. x ( x = y -> ph ) is a way of expressing " y substituted for x in wff ph " (cf. sb6 1858). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1799, ax11v2 1792 and ax-11o 1795. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Axiom	ax-i12 1485	Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. This axiom has been modified from the original ax-12 1489 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Axiom	ax-bndl 1486	Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as A. z z = x \/ -. A. z z = x. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object. A. z z = x holds if z and x are the same variable, likewise for z and y, and A. x A. z ( x = y -> A. z x = y ) holds if z is distinct from the others (and the universe has at least two objects). As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1485 as can be seen at axi12 1494. Whether ax-bndl 1486 can be proved from the remaining axioms including ax-i12 1485 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)
|- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
Axiom	ax-4 1487	Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for ph). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only x is free in A. y x = y.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1425. Conditional forms of the converse are given by ax-12 1489, ax-16 1786, and ax-17 1506. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1748. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> ph )
Theorem	sp 1488	Specialization. Another name for ax-4 1487. (Contributed by NM, 21-May-2008.)
|- ( A. x ph -> ph )
Theorem	ax-12 1489	Rederive the original version of the axiom from ax-i12 1485. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
Theorem	ax12or 1490	Another name for ax-i12 1485. (Contributed by NM, 3-Feb-2015.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Axiom	ax-13 1491	Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the e. binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z -> y e. z ) )
Axiom	ax-14 1492	Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the e. binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	hbequid 1493	Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1423, ax-8 1482, ax-12 1489, and ax-gen 1425. This shows that this can be proved without ax-9 1511, even though the theorem equid 1677 cannot be. A shorter proof using ax-9 1511 is obtainable from equid 1677 and hbth 1439.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
|- ( x = x -> A. y x = x )
Theorem	axi12 1494	Proof that ax-i12 1485 follows from ax-bndl 1486. So that we can track which theorems rely on ax-bndl 1486, proofs should reference ax-i12 1485 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Theorem	alequcom 1495	Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> A. y y = x )
Theorem	alequcoms 1496	A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> ph )   =>    |- ( A. y y = x -> ph )
Theorem	nalequcoms 1497	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
|- ( -. A. x x = y -> ph )   =>    |- ( -. A. y y = x -> ph )
Theorem	nfr 1498	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
|- ( F/ x ph -> ( ph -> A. x ph ) )
Theorem	nfri 1499	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- ( ph -> A. x ph )
Theorem	nfrd 1500	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> ( ps -> A. x ps ) )