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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremretbwax1 1801 tbw-ax1 1766 rederived from merco1 1779.

This theorem, along with retbwax2 1782, retbwax3 1789, and retbwax4 1781, shows that merco1 1779 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.)

((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom

Theoremmerco2 1802 A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1779. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏 → (𝜂𝜑))))

Theoremmercolem1 1803 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))

Theoremmercolem2 1804 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → (𝜒 → (𝜃𝜑)))

Theoremmercolem3 1805 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒) → (𝜓 → (𝜑𝜒)))

Theoremmercolem4 1806 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))

Theoremmercolem5 1807 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → ((𝜃𝜑) → (𝜏 → (𝜒𝜑))))

Theoremmercolem6 1808 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))

Theoremmercolem7 1809 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))

Theoremmercolem8 1810 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))

Theoremre1tbw1 1811 tbw-ax1 1766 rederived from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremre1tbw2 1812 tbw-ax2 1767 rederived from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremre1tbw3 1813 tbw-ax3 1768 rederived from merco2 1802. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremre1tbw4 1814 tbw-ax4 1769 rederived from merco2 1802.

This theorem, along with re1tbw1 1811, re1tbw2 1812, and re1tbw3 1813, shows that merco2 1802, 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.)

(⊥ → 𝜑)

1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms

Theoremrb-bijust 1815 Justification for rb-imdf 1816. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))

Theoremrb-imdf 1816 The definition of implication, in terms of and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ (𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ¬ (¬ (¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremanmp 1817 Modus ponens for ¬ axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜑𝜓)       𝜓

Theoremrb-ax1 1818 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.)
(¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))

Theoremrb-ax2 1819 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.)
(¬ (𝜑𝜓) ∨ (𝜓𝜑))

Theoremrb-ax3 1820 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.)
𝜑 ∨ (𝜓𝜑))

Theoremrb-ax4 1821 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.)
(¬ (𝜑𝜑) ∨ 𝜑)

Theoremrbsyl 1822 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓𝜒)    &   (𝜑𝜓)       (𝜑𝜒)

Theoremrblem1 1823 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜓)    &   𝜒𝜃)       (¬ (𝜑𝜒) ∨ (𝜓𝜃))

Theoremrblem2 1824 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ (𝜒 ∨ (𝜑𝜓)))

Theoremrblem3 1825 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ ((𝜒𝜓) ∨ 𝜑))

Theoremrblem4 1826 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜃)    &   𝜓𝜏)    &   𝜒𝜂)       (¬ ((𝜑𝜓) ∨ 𝜒) ∨ ((𝜂𝜏) ∨ 𝜃))

Theoremrblem5 1827 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (¬ ¬ 𝜑𝜓) ∨ (¬ ¬ 𝜓𝜑))

Theoremrblem6 1828 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜑𝜓)

Theoremrblem7 1829 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜓𝜑)

Theoremre1axmp 1830 ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoremre2luk1 1831 luk-1 1721 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremre2luk2 1832 luk-2 1722 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremre2luk3 1833 luk-3 1723 derived from Russell-Bernays'.

This theorem, along with re1axmp 1830, re2luk1 1831, and re2luk2 1832 shows that rb-ax1 1818, rb-ax2 1819, rb-ax3 1820, and rb-ax4 1821, along with anmp 1817, 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.)

(𝜑 → (¬ 𝜑𝜓))

1.3.11  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 188, modus ponendo tollens I mptnan 1834, modus ponendo tollens II mptxor 1835, and modus tollendo ponens (exclusive-or version) mtpxor 1837. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1837 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 1836.

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 1838 and stoic1b 1839) and thema 3 (stoic3 1842). 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 1840, stoic2b 1841, stoic4a 1843, and stoic4b 1844.

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, (𝜑 ↔ ¬ ¬ 𝜑) as proven in notnotb 304.

"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, (𝜑𝜑) 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, (𝜑 ⊻ ¬ 𝜑) as proven in xorexmid 1621.

[Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1608 and ancom 465) and the principle of contraposition (con4 112) (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 2693 and related theorems.

Theoremmptnan 1834 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 1835) 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.)
𝜑    &    ¬ (𝜑𝜓)        ¬ 𝜓

Theoremmptxor 1835 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.)
𝜑    &   (𝜑𝜓)        ¬ 𝜓

Theoremmtpor 1836 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1837, 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 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 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.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremmtpxor 1837 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1836, one of the five "indemonstrables" in Stoic logic. The rule says, "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1836. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1835, that is, it is exclusive-or df-xor 1606), 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 1835), 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.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremstoic1a 1838 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 1838 and stoic1b 1839 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)

((𝜑𝜓) → 𝜃)       ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

Theoremstoic1b 1839 Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1838. (Contributed by David A. Wheeler, 16-Feb-2019.)
((𝜑𝜓) → 𝜃)       ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)

Theoremstoic2a 1840 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 𝜑, 𝜓𝜒; in Metamath we will represent that construct as 𝜑𝜓𝜒. This version a is without the phrase "or both"; see stoic2b 1841 for the version with the phrase "or both". We already have this rule as syldan 488, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic2b 1841 Stoic logic Thema 2 version b. See stoic2a 1840. Version b is with the phrase "or both". We already have this rule as mpd3an3 1566, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic3 1842 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.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4a 1843 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 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1844 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜒𝜑𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4b 1844 Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1843 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   (((𝜒𝜑𝜓) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

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 1878) 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 1846) 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 2192) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 2110 below).

Theorem spw 2110 can be used to prove any instance of sp 2192 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2192 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2192 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 2192 as theorem axc5 34674 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 1863, ax-4 1878, ax-5 1980, ax-6 2046, ax-7 2082, ax-8 2133, and ax-9 2140. The last three are equality axioms that represent three sub-schemes of Tarski's scheme B8. Due to its side-condition ("where 𝜑 is an atomic formula and 𝜓 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 2082, ax-8 2133, and ax-9 2140 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 2192, even though (using spw 2110) 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 2160, ax-11 2175, ax-12 2188, and ax-13 2383 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "scheme 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 1622 for use by df-tru 1627. See the comments in that section. In this section, we continue with the first "real" use of it.

1.4.1.1  Existential quantifier

Syntaxwex 1845 Extend wff definition to include the existential quantifier ("there exists").
wff 𝑥𝜑

Definitiondf-ex 1846 Define existential quantification. 𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true." Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
(∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)

Theoremalnex 1847 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)

Theoremeximal 1848 A utility theorem. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also alimex 1899. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

1.4.1.2  Non-freeness predicate

Syntaxwnf 1849 Extend wff definition to include the not-free predicate.
wff 𝑥𝜑

SyntaxwnfOLD 1850 Extend wff definition to include the old not-free predicate. Obsolete as of 16-Sep-2021. (New usage is discouraged.)
wff 𝑥𝜑

Definitiondf-nf 1851 Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 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 2509). An example of where this is used is stdpc5 2215. See nf5 2255 for an alternate definition which involves 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 formula 𝑥 = 𝑥 (see nfequid 2087), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the formula 𝑥 = 𝑥 cannot affect the truth of that formula (and thus substitutions will not change the result).

This definition of not-free tightly ties to the quantifier 𝑥. At this state (no axioms restricting quantifiers yet) 'non-free' appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization.

The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 2050.

This predicate only applies to wffs. See df-nfc 2883 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Converted to definition. (Revised by BJ, 6-May-2019.)

(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremnf2 1852 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Theoremnf3 1853 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Theoremnf4 1854 Alternate definition of non-freeness. This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Theoremnfi 1855 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremnfri 1856 Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
𝑥𝜑       (∃𝑥𝜑 → ∀𝑥𝜑)

Theoremnfd 1857 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
(𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnfrd 1858 Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Theoremnftht 1859 Closed form of nfth 1868. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.)
(∀𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremnfntht 1860 Closed form of nfnth 1869. (Contributed by BJ, 16-Sep-2021.)
(¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremnfntht2 1861 Closed form of nfnth 1869. (Contributed by BJ, 16-Sep-2021.)
(∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Definitiondf-nfOLD 1862 Obsolete definition replaced by nf5 2255 as of 3-Oct-2021. This definition is less suitable than df-nf 1851 when ax-10 2160 and ax-12 2188 are not in effect. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

1.4.2  Rule scheme ax-gen (Generalization)

Axiomax-gen 1863 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 𝑥𝑥 = 𝑥 or even 𝑦𝑥 = 𝑥. Theorem allt 32698 shows the special case 𝑥. Theorem spi 2193 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.)
𝜑       𝑥𝜑

Theoremgen2 1864 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
𝜑       𝑥𝑦𝜑

Theoremmpg 1865 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓

Theoremmpgbi 1866 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓

Theoremmpgbir 1867 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(𝜑 ↔ ∀𝑥𝜓)    &   𝜓       𝜑

Theoremnfth 1868 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1851 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
𝜑       𝑥𝜑

Theoremnfnth 1869 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1851 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
¬ 𝜑       𝑥𝜑

Theoremhbth 1870 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 (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 11-May-1993.)

𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnftru 1871 The true constant has no free variables. (This can also be proven in one step with nfv 1984, but this proof does not use ax-5 1980.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥

Theoremnex 1872 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑

Theoremnffal 1873 The false constant has no free variables (see nftru 1871). (Contributed by BJ, 6-May-2019.)
𝑥

Theoremsptruw 1874 Version of sp 2192 when 𝜑 is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
𝜑       (∀𝑥𝜑𝜑)

TheoremnfiOLD 1875 Obsolete proof of nf5i 2165 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑

TheoremnfthOLD 1876 Obsolete proof of nfth 1868 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝑥𝜑

TheoremnfnthOLD 1877 Obsolete proof of nfnth 1869 as of 6-Oct-2021. (Contributed by Mario Carneiro, 6-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝜑       𝑥𝜑

1.4.3  Axiom scheme ax-4 (Quantified Implication)

Axiomax-4 1878 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1879 for labeling consistency. It should be used only by alim 1879. (Contributed by NM, 21-May-2008.) Use alim 1879 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalim 1879 Restatement of Axiom ax-4 1878, for labeling consistency. It should be the only theorem using ax-4 1878. (Contributed by NM, 10-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalimi 1880 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theorem2alimi 1881 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)

Theoremala1 1882 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥𝜑 → ∀𝑥(𝜓𝜑))

Theoremal2im 1883 Closed form of al2imi 1884. Version of alim 1879 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Theoremal2imi 1884 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalanimi 1885 Variant of al2imi 1884 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Theoremalimdh 1886 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1879. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalbi 1887 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Theoremalbii 1888 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Theorem2albii 1889 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓)

Theoremsylgt 1890 Closed form of sylg 1891. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))

Theoremsylg 1891 A syllogism combined with generalization. Inference associated with sylgt 1890. General form of alrimih 1892. (Contributed by BJ, 4-Oct-2019.)
(𝜑 → ∀𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝜒)

Theoremalrimih 1892 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2214 and 19.21h 2260. Instance of sylg 1891. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremhbxfrbi 1893 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2860 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑 → ∀𝑥𝜑)

Theoremalex 1894 Universal quantifier in terms of existential quantifier and negation. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Theoremexnal 1895 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)

Theorem2nalexn 1896 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)

Theorem2exnaln 1897 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Theorem2nexaln 1898 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremalimex 1899 A utility theorem. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also eximal 1848. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Theoremaleximi 1900 A variant of al2imi 1884: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2169, sps 2194 and eximd 2224, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

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