P. 17 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rblem5 1601	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( -. -. ph \/ ps ) \/ ( -. -. ps \/ ph ) )
Theorem	rblem6 1602	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) )   =>    |- ( -. ph \/ ps )
Theorem	rblem7 1603	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) )   =>    |- ( -. ps \/ ph )
Theorem	re1axmp 1604	ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ph   &    |- ( ph -> ps )   =>    |- ps
Theorem	re2luk1 1605	luk-1 1495 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
Theorem	re2luk2 1606	luk-2 1496 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( -. ph -> ph ) -> ph )
Theorem	re2luk3 1607	luk-3 1497 derived from Russell-Bernays'. This theorem, along with re1axmp 1604, re2luk1 1605, and re2luk2 1606 shows that rb-ax1 1592, rb-ax2 1593, rb-ax3 1594, and rb-ax4 1595, along with anmp 1591, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> ( -. ph -> ps ) )
1.3.10  Stoic logic non-modal portion (Chrysippus of Soli) The Greek Stoics developed a system of logic called Stoic 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). In this section we show that the propositional logic system we use (which is non-modal) is at least as strong as the non-modal portion of Stoic logic. We show this by showing that our system assumes or proves all of key features of Stoic logic's non-modal portion (specifically the Stoic logic indemonstrables, themata, and principles). "In terms of contemporary logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural-deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata. An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of the themata (Diogenes Laertius (D. L. 7.78))." (Ancient Logic, Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/logic-ancient/). There are also a few "principles" that support logical reasoning, discussed below. For more information, see "Stoic Logic" by Susanne Bobzien, especially [Bobzien] p. 110-120, especially for a discussion about the themata (including how they were reconstructed and how they were used). There are differences in the systems we can only partly represent, for example, in Stoic logic "truth and falsehood are temporal properties of assertibles... They can belong to an assertible at one time but not at another" ([Bobzien] p. 87). Stoic logic also included various kinds of modalities, which we do not include here since our basic propositional logic does not include modalities. 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 176, modus ponendo tollens I mptnan 1608, modus ponendo tollens II mptxor 1609, and modus tollendo ponens (exclusive-or version) mtpxor 1611. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1611 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 1610. After we prove the indemonstratables, we then prove all the Stoic logic themata (the inference rules of Stoic logic; "thema" is singular). This is straightforward for thema 1 (stoic1a 1612 and stoic1b 1614) and thema 3 (stoic3 1617). However, while Stoic logic was once a leading logic system, most direct information about Stoic logic has since been lost, including the exact texts of thema 2 and thema 4. There are, however, enough references and specific examples to support reconstruction. Themata 2 and 4 have been reconstructed; see statements T2 and T4 in [Bobzien] p. 110-120 and our proofs of them in stoic2a 1615, stoic2b 1616, stoic4a 1618, and stoic4b 1619. Stoic logic also had a set of principles involving assertibles. Statements in [Bobzien] p. 99 express the known principles. The following paragraphs discuss these principles and our proofs of them. "A principle of double negation, expressed by saying that a double-negation (Not: not: p) is equivalent to the assertible that is doubly negated (p) (DL VII 69)." In other words, ( ph <-> -. -. ph ) as proven in notnot 289. "The principle that all conditionals that are formed by using the same assertible twice (like 'If p, p') are true (Cic. Acad. II 98)." In other words, ( ph -> ph ) as proven in id 22. "The principle that all disjunctions formed by a contradiction (like 'Either p or not: p') are true (S. E. M VIII 282)" Remember that in Stoic logic, 'or' means 'exclusive or'. In other words, ( ph \/_ -. ph ) as proven in xorexmid 1381. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1365 and ancom 448) and the principle of contraposition (ax-3 8) (pointing to DL VII 194). In short, the non-modal propositional logic system we use is at least as strong as the non-modal portion of Stoic logic. For more about Aristotle's system, see barbara 2321 and related theorems.
Theorem	mptnan 1608	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 1609) 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 1609	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, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
|- ph   &    |- ( ph \/_ ps )   =>    |- -. ps
Theorem	mtpor 1610	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1611, 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 alternative 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 1611	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1610, 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 1610. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1609, that is, it is exclusive-or df-xor 1363), 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 1609), 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 1612	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 1612 and stoic1b 1614 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	stoic1aOLD 1613	Obsolete proof of stoic1a 1612 as of 20-May-2020. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ph /\ -. th ) -> -. ps )
Theorem	stoic1b 1614	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1612. (Contributed by David A. Wheeler, 16-Feb-2019.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ps /\ -. th ) -> -. ph )
Theorem	stoic2a 1615	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 1616 for the version with the phrase "or both". We already have this rule as syldan 468, 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 1616	Stoic logic Thema 2 version b. See stoic2a 1615. Version b is with the phrase "or both". We already have this rule as mpd3an3 1323, 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 1617	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 1618	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 1619 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 1619	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1618 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ( ch /\ ph /\ ps ) /\ th ) -> ta )   =>    |- ( ( ph /\ ps /\ th ) -> ta )
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes) Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all", e.g. ax-4 1639) in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1621) which is defined in terms of universal quantification. Our axioms are really axiom schemes, and our wff and setvar variables are metavariables ranging over expressions in an underlying "object language." This is explained here: mmset.html#axiomnote. Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 1867) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1815 below). Theorem spw 1815 can be used to prove any instance of sp 1867 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 1867 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1867 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 1867 as theorem axc5 35037 using the auxiliary axioms that make our system metalogically complete. Our version of Tarski's system S2 consists of propositional calculus (ax-mp 5, ax-1 6, ax-2 7, ax-3 8) plus ax-gen 1626, ax-4 1639, ax-5 1712, ax-6 1755, ax-7 1798, ax-8 1828, and ax-9 1830. The last 3 are equality axioms that represent 3 sub-schemes of Tarski's scheme B8. Due to its side-condition ("where ph is an atomic formula and ps is obtained by replacing an occurrence of the variable x by the variable y"), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-7 1798, ax-8 1828, and ax-9 1830 are sufficient for set theory and much easier to work with. Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of "free variable" and "proper substitution") is arguably easier for a non-logician human to follow step by step in a proof (where "follow" means being able to identify the substitutions that were made, without necessarily a higher-level understanding). In particular, it is logically complete in that it can derive all possible object-language theorems of predicate calculus with equality, i.e. the same theorems as the traditional system can derive. However, for efficiency (and indeed a key feature that makes Metamath successful), our system is designed to derive reusable theorem schemes (rather than object-language theorems) from other schemes. From this "metalogical" point of view, Tarski's S2 is not complete. For example, we cannot derive scheme sp 1867, even though (using spw 1815) we can derive all instances of it that don't involve wff metavariables or bundled set metavariables. (Two set metavariables are "bundled" if they can be substituted with the same set metavariable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-10 1845, ax-11 1850, ax-12 1862, and ax-13 2006 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "metalogical completeness," allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language.
1.4.1  Universal quantifier (continued); define "exists" and "not free" The universal quantifier was introduced above in wal 1397 for use by df-tru 1402. See the comments in that section. In this section, we continue with the first "real" use of it.
Syntax	wex 1620	Extend wff definition to include the existential quantifier ("there exists").
wff E. x ph
Definition	df-ex 1621	Define existential quantification. E. x ph means "there exists at least one set x such that ph is true." Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
|- ( E. x ph <-> -. A. x -. ph )
Theorem	alnex 1622	Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
|- ( A. x -. ph <-> -. E. x ph )
Theorem	eximal 1623	A utility theorem. An interesting case is when the same formula is substituted for both ph and ps, since then both implications express a type of non-freeness. See also alimex 1660. (Contributed by BJ, 12-May-2019.)
|- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) )
Syntax	wnf 1624	Extend wff definition to include the not-free predicate.
wff F/ x ph
Definition	df-nf 1625	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 2125). An example of where this is used is stdpc5 1916. See nf2 1968 for an alternative 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 (see nfequid 1800), 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). This predicate only applies to wffs. See df-nfc 2532 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
1.4.2  Rule scheme ax-gen (Generalization)
Axiom	ax-gen 1626	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 allt 30019 shows the special case A. x T.. Theorem spi 1872 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, 3-Jan-1993.)
|- ph   =>    |- A. x ph
Theorem	gen2 1627	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
|- ph   =>    |- A. x A. y ph
Theorem	mpg 1628	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
|- ( A. x ph -> ps )   &    |- ph   =>    |- ps
Theorem	mpgbi 1629	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 1630	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	nfi 1631	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 1632	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, 11-May-1993.)
|- ph   =>    |- ( ph -> A. x ph )
Theorem	nfth 1633	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- F/ x ph
Theorem	nftru 1634	The true constant has no free variables. (This can also be proven in one step with nfv 1715, but this proof does not use ax-5 1712.) (Contributed by Mario Carneiro, 6-Oct-2016.)
|- F/ x T.
Theorem	nex 1635	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
|- -. ph   =>    |- -. E. x ph
Theorem	nfnth 1636	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
|- -. ph   =>    |- F/ x ph
Theorem	nffal 1637	The false constant has no free variables (see nftru 1634). (Contributed by BJ, 6-May-2019.)
|- F/ x F.
Theorem	sptruw 1638	Version of sp 1867 when ph is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
|- ph   =>    |- ( A. x ph -> ph )
1.4.3  Axiom scheme ax-4 (Quantified Implication)
Axiom	ax-4 1639	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1640 for labelling consistency. It should be used only by alim 1640. (Contributed by NM, 21-May-2008.) (New usage is discouraged.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	alim 1640	Restatement of Axiom ax-4 1639, for labelling consistency. It should be the only theorem using ax-4 1639. (Contributed by NM, 10-Jan-1993.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	alimi 1641	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	2alimi 1642	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	al2im 1643	Closed form of al2imi 1644. Version of ax-4 1639 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( A. x ( ph -> ( ps -> ch ) ) -> ( A. x ph -> ( A. x ps -> A. x ch ) ) )
Theorem	al2imi 1644	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x ph -> ( A. x ps -> A. x ch ) )
Theorem	alanimi 1645	Variant of al2imi 1644 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x ph /\ A. x ps ) -> A. x ch )
Theorem	alimdh 1646	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1640. (Contributed by NM, 4-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	albi 1647	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
|- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) )
Theorem	albii 1648	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 1649	Inference adding two 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	alrimih 1650	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 1913 and 19.21h 1915. (Contributed by NM, 9-Jan-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	hbxfrbi 1651	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2504 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ps )   &    |- ( ps -> A. x ps )   =>    |- ( ph -> A. x ph )
Theorem	nfbii 1652	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   =>    |- ( F/ x ph <-> F/ x ps )
Theorem	nfxfr 1653	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 1654	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	alex 1655	Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
|- ( A. x ph <-> -. E. x -. ph )
Theorem	exnal 1656	Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( E. x -. ph <-> -. A. x ph )
Theorem	2nalexn 1657	Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( -. A. x A. y ph <-> E. x E. y -. ph )
Theorem	2exnaln 1658	Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ph <-> -. A. x A. y -. ph )
Theorem	2nexaln 1659	Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( -. E. x E. y ph <-> A. x A. y -. ph )
Theorem	alimex 1660	A utility theorem. An interesting case is when the same formula is substituted for both ph and ps, since then both implications express a type of non-freeness. See also eximal 1623. (Contributed by BJ, 12-May-2019.)
|- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) )
Theorem	aleximi 1661	A variant of al2imi 1644: instead of applying A. x quantifiers to the final implication, replace them with E. x. A shorter proof is possible using nfa1 1905, sps 1873 and eximd 1890, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x ph -> ( E. x ps -> E. x ch ) )
Theorem	exim 1662	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
Theorem	eximOLD 1663	Obsolete proof of exim 1662 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
Theorem	eximi 1664	Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
|- ( ph -> ps )   =>    |- ( E. x ph -> E. x ps )
Theorem	2eximi 1665	Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> ps )   =>    |- ( E. x E. y ph -> E. x E. y ps )
Theorem	eximii 1666	Inference associated with eximi 1664. (Contributed by BJ, 3-Feb-2018.)
|- E. x ph   &    |- ( ph -> ps )   =>    |- E. x ps
Theorem	aleximiOLD 1667	Obsolete proof of aleximi 1661 as of 4-Sep-2019. (Contributed by Wolf Lammen, 18-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x ph -> ( E. x ps -> E. x ch ) )
Theorem	ala1 1668	Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
|- ( A. x ph -> A. x ( ps -> ph ) )
Theorem	exa1 1669	Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
|- ( E. x ph -> E. x ( ps -> ph ) )
Theorem	19.38 1670	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 1912 and 19.23t 1917. (Revised by Wolf Lammen, 2-Jan-2018.)
|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
Theorem	alinexa 1671	A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
|- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) )
Theorem	alexn 1672	A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
|- ( A. x E. y -. ph <-> -. E. x A. y ph )
Theorem	2exnexn 1673	Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
|- ( E. x A. y ph <-> -. A. x E. y -. ph )
Theorem	exbi 1674	Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) )
Theorem	exbii 1675	Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
|- ( ph <-> ps )   =>    |- ( E. x ph <-> E. x ps )
Theorem	2exbii 1676	Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
|- ( ph <-> ps )   =>    |- ( E. x E. y ph <-> E. x E. y ps )
Theorem	3exbii 1677	Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
|- ( ph <-> ps )   =>    |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )
Theorem	exanali 1678	A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
|- ( E. x ( ph /\ -. ps ) <-> -. A. x ( ph -> ps ) )
Theorem	exancom 1679	Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
|- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
Theorem	alrimdh 1680	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1913 and 19.21h 1915. (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	eximdh 1681	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	nexdh 1682	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> -. ps )   =>    |- ( ph -> -. E. x ps )
Theorem	albidh 1683	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	exbidh 1684	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	exsimpl 1685	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( E. x ( ph /\ ps ) -> E. x ph )
Theorem	exsimpr 1686	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( E. x ( ph /\ ps ) -> E. x ps )
Theorem	19.40 1687	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
|- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
Theorem	19.26 1688	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
Theorem	19.26-2 1689	Theorem 19.26 1688 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 1690	Theorem 19.26 1688 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
|- ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )
Theorem	19.29 1691	Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1692. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r 1692	Variation of 19.29 1691. (Contributed by NM, 18-Aug-1993.)
|- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r2 1693	Variation of 19.29r 1692 with double quantification. (Contributed by NM, 3-Feb-2005.)
|- ( ( E. x E. y ph /\ A. x A. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.29x 1694	Variation of 19.29 1691 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
|- ( ( E. x A. y ph /\ A. x E. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.35 1695	Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
|- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
Theorem	19.35OLD 1696	Obsolete proof of 19.35 1695 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
Theorem	19.35i 1697	Inference associated with 19.35 1695. (Contributed by NM, 21-Jun-1993.)
|- E. x ( ph -> ps )   =>    |- ( A. x ph -> E. x ps )
Theorem	19.35ri 1698	Inference associated with 19.35 1695. (Contributed by NM, 12-Mar-1993.)
|- ( A. x ph -> E. x ps )   =>    |- E. x ( ph -> ps )
Theorem	19.25 1699	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( A. y E. x ( ph -> ps ) -> ( E. y A. x ph -> E. y E. x ps ) )
Theorem	19.30 1700	Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )