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	merco1lem8 1601	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> ( ( ps -> ( ps -> ch ) ) -> ( ps -> ch ) ) )
Theorem	merco1lem9 1602	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )
Theorem	merco1lem10 1603	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) -> ( th -> ph ) )
Theorem	merco1lem11 1604	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> F. ) -> ps ) )
Theorem	merco1lem12 1605	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps ) )
Theorem	merco1lem13 1606	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph -> ta ) )
Theorem	merco1lem14 1607	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) )
Theorem	merco1lem15 1608	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) )
Theorem	merco1lem16 1609	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ph -> ( ps -> ch ) ) -> ta ) -> ( ( ph -> ch ) -> ta ) )
Theorem	merco1lem17 1610	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ( ( ph -> ps ) -> ph ) -> ch ) -> ta ) -> ( ( ph -> ch ) -> ta ) )
Theorem	merco1lem18 1611	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1590. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ps -> ph ) -> ( ps -> ch ) ) )
Theorem	retbwax1 1612	tbw-ax1 1577 rederived from merco1 1590. This theorem, along with retbwax2 1593, retbwax3 1600, and retbwax4 1592, shows that merco1 1590 with ax-mp 5 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
Theorem	merco2 1613	A single axiom for propositional calculus offered by Meredith. This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1590. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ( et -> ph ) ) ) )
Theorem	mercolem1 1614	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) )
Theorem	mercolem2 1615	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ph -> ps ) -> ph ) -> ( ch -> ( th -> ph ) ) )
Theorem	mercolem3 1616	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ps -> ch ) -> ( ps -> ( ph -> ch ) ) )
Theorem	mercolem4 1617	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( th -> ( et -> ph ) ) -> ( ( ( th -> ch ) -> ph ) -> ( ta -> ( et -> ph ) ) ) )
Theorem	mercolem5 1618	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) )
Theorem	mercolem6 1619	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ( ps -> ( ph -> ch ) ) ) -> ( ps -> ( ph -> ch ) ) )
Theorem	mercolem7 1620	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ( th -> ps ) ) -> ( th -> ps ) ) )
Theorem	mercolem8 1621	Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( ta -> ( th -> ( ph -> ch ) ) ) ) )
Theorem	re1tbw1 1622	tbw-ax1 1577 rederived from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
Theorem	re1tbw2 1623	tbw-ax2 1578 rederived from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> ( ps -> ph ) )
Theorem	re1tbw3 1624	tbw-ax3 1579 rederived from merco2 1613. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ( ph -> ps ) -> ph ) -> ph )
Theorem	re1tbw4 1625	tbw-ax4 1580 rederived from merco2 1613. This theorem, along with re1tbw1 1622, re1tbw2 1623, and re1tbw3 1624, shows that merco2 1613, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( F. -> ph )
1.3.9  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms
Theorem	rb-bijust 1626	Justification for rb-imdf 1627. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( ph <-> ps ) <-> -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) )
Theorem	rb-imdf 1627	The definition of implication, in terms of \/ and -.. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ( -. ( -. ( ph -> ps ) \/ ( -. ph \/ ps ) ) \/ -. ( -. ( -. ph \/ ps ) \/ ( ph -> ps ) ) )
Theorem	anmp 1628	Modus ponens for \/ -. axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ph   &    |- ( -. ph \/ ps )   =>    |- ps
Theorem	rb-ax1 1629	The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( -. ps \/ ch ) \/ ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) )
Theorem	rb-ax2 1630	The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( ph \/ ps ) \/ ( ps \/ ph ) )
Theorem	rb-ax3 1631	The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ph \/ ( ps \/ ph ) )
Theorem	rb-ax4 1632	The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( ph \/ ph ) \/ ph )
Theorem	rbsyl 1633	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ps \/ ch )   &    |- ( ph \/ ps )   =>    |- ( ph \/ ch )
Theorem	rblem1 1634	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ph \/ ps )   &    |- ( -. ch \/ th )   =>    |- ( -. ( ph \/ ch ) \/ ( ps \/ th ) )
Theorem	rblem2 1635	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( ch \/ ph ) \/ ( ch \/ ( ph \/ ps ) ) )
Theorem	rblem3 1636	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ( ch \/ ph ) \/ ( ( ch \/ ps ) \/ ph ) )
Theorem	rblem4 1637	Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. ph \/ th )   &    |- ( -. ps \/ ta )   &    |- ( -. ch \/ et )   =>    |- ( -. ( ( ph \/ ps ) \/ ch ) \/ ( ( et \/ ta ) \/ th ) )
Theorem	rblem5 1638	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 1639	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 1640	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 1641	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 1642	luk-1 1532 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 1643	luk-2 1533 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( -. ph -> ph ) -> ph )
Theorem	re2luk3 1644	luk-3 1534 derived from Russell-Bernays'. This theorem, along with re1axmp 1641, re2luk1 1642, and re2luk2 1643 shows that rb-ax1 1629, rb-ax2 1630, rb-ax3 1631, and rb-ax4 1632, along with anmp 1628, 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 179, modus ponendo tollens I mptnan 1645, modus ponendo tollens II mptxor 1646, and modus tollendo ponens (exclusive-or version) mtpxor 1648. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1648 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 1647. 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 1649 and stoic1b 1651) and thema 3 (stoic3 1654). 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 1652, stoic2b 1653, stoic4a 1655, and stoic4b 1656. 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 292. "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 1419. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1403 and ancom 451) 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 2362 and related theorems.
Theorem	mptnan 1645	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 1646) 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 1646	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 1647	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1648, 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 1648	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1647, 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 1647. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1646, that is, it is exclusive-or df-xor 1401), 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 1646), 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 1649	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 1649 and stoic1b 1651 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 1650	Obsolete proof of stoic1a 1649 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 1651	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1649. (Contributed by David A. Wheeler, 16-Feb-2019.)
|- ( ( ph /\ ps ) -> th )   =>    |- ( ( ps /\ -. th ) -> -. ph )
Theorem	stoic2a 1652	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 1653 for the version with the phrase "or both". We already have this rule as syldan 472, 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 1653	Stoic logic Thema 2 version b. See stoic2a 1652. Version b is with the phrase "or both". We already have this rule as mpd3an3 1361, 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 1654	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 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 1655	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 1656 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 1656	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1655 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 1676) 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 1658) 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 1914) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1861 below). Theorem spw 1861 can be used to prove any instance of sp 1914 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 1914 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1914 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 1914 as theorem axc5 32434 using the auxiliary axiom schemes 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 1663, ax-4 1676, ax-5 1752, ax-6 1798, ax-7 1843, ax-8 1874, and ax-9 1876. The last three are equality axioms that represent three 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 1843, ax-8 1874, and ax-9 1876 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 1914, even though (using spw 1861) we can derive all instances of it that don't involve wff metavariables or bundled setvar variables. (Two setvar variables are "bundled" if they can be substituted with the same setvar variable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-10 1891, ax-11 1896, ax-12 1909, and ax-13 2057 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 1435 for use by df-tru 1440. See the comments in that section. In this section, we continue with the first "real" use of it.
Syntax	wex 1657	Extend wff definition to include the existential quantifier ("there exists").
wff E. x ph
Definition	df-ex 1658	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 1659	Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
|- ( A. x -. ph <-> -. E. x ph )
Theorem	eximal 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 alimex 1697. (Contributed by BJ, 12-May-2019.)
|- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) )
Syntax	wnf 1661	Extend wff definition to include the not-free predicate.
wff F/ x ph
Definition	df-nf 1662	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 2178). An example of where this is used is stdpc5 1967. See nf2 2020 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 1846), 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 2568 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 1663	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 31066 shows the special case A. x T.. Theorem spi 1919 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 1664	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
|- ph   =>    |- A. x A. y ph
Theorem	mpg 1665	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
|- ( A. x ph -> ps )   &    |- ph   =>    |- ps
Theorem	mpgbi 1666	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 1667	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 1668	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 1669	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 1670	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- F/ x ph
Theorem	nftru 1671	The true constant has no free variables. (This can also be proven in one step with nfv 1755, but this proof does not use ax-5 1752.) (Contributed by Mario Carneiro, 6-Oct-2016.)
|- F/ x T.
Theorem	nex 1672	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
|- -. ph   =>    |- -. E. x ph
Theorem	nfnth 1673	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
|- -. ph   =>    |- F/ x ph
Theorem	nffal 1674	The false constant has no free variables (see nftru 1671). (Contributed by BJ, 6-May-2019.)
|- F/ x F.
Theorem	sptruw 1675	Version of sp 1914 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 1676	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1677 for labeling consistency. It should be used only by alim 1677. (Contributed by NM, 21-May-2008.) (New usage is discouraged.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	alim 1677	Restatement of Axiom ax-4 1676, for labeling consistency. It should be the only theorem using ax-4 1676. (Contributed by NM, 10-Jan-1993.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	alimi 1678	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	2alimi 1679	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 1680	Closed form of al2imi 1681. Version of ax-4 1676 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 1681	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 1682	Variant of al2imi 1681 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x ph /\ A. x ps ) -> A. x ch )
Theorem	alimdh 1683	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1677. (Contributed by NM, 4-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	albi 1684	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 1685	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 1686	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 1687	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 1964 and 19.21h 1966. (Contributed by NM, 9-Jan-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	hbxfrbi 1688	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2539 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ps )   &    |- ( ps -> A. x ps )   =>    |- ( ph -> A. x ph )
Theorem	nfbii 1689	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   =>    |- ( F/ x ph <-> F/ x ps )
Theorem	nfxfr 1690	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 1691	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 1692	Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
|- ( A. x ph <-> -. E. x -. ph )
Theorem	exnal 1693	Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
|- ( E. x -. ph <-> -. A. x ph )
Theorem	2nalexn 1694	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 1695	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 1696	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 1697	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 1660. (Contributed by BJ, 12-May-2019.)
|- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) )
Theorem	aleximi 1698	A variant of al2imi 1681: instead of applying A. x quantifiers to the final implication, replace them with E. x. A shorter proof is possible using nfa1 1956, sps 1920 and eximd 1937, 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	alexbii 1699	Biconditional form of aleximi 1698. (Contributed by BJ, 16-Nov-2020.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( A. x ph -> ( E. x ps <-> E. x ch ) )
Theorem	exim 1700	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 ) )