P. 19 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
1.3.12  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 200, modus ponendo tollens I mptnan 1801, modus ponendo tollens II mptxor 1802, and modus tollendo ponens (exclusive-or version) mtpxor 1804. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1804 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 1803. 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 1805 and stoic1b 1806) and thema 3 (stoic3 1809). 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 1807, stoic2b 1808, stoic4a 1810, and stoic4b 1811. 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 318. "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, (𝜑 → 𝜑) as proven in id 23. "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 1557. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1544 and ancom 466) and the principle of contraposition (con4 114) (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.
Theorem	mptnan 1801	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 1802) 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    &   ⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mptxor 1802	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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mtpor 1803	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1804, 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 alternate phrasing is: "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth". -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
⊢ ¬ 𝜑    &   ⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mtpxor 1804	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1803, 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 1803. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1802, that is, it is exclusive-or df-xor 1542), 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 1802), 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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ 𝜓
Theorem	stoic1a 1805	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 1805 and stoic1b 1806 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
Theorem	stoic1b 1806	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1805. (Contributed by David A. Wheeler, 16-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)
Theorem	stoic2a 1807	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, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1808 for the version with the phrase "or both". We already have this rule as syldan 603, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic2b 1808	Stoic logic Thema 2 version b. See stoic2a 1807. Version b is with the phrase "or both". We already have this rule as mpd3an3 1491, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic3 1809	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 that 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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4a 1810	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 1811 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4b 1811	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1810 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 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 to talk about individual objects in a domain of discourse (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 1842) in order to make statements about whether a wff holds for every object in the domain of discourse. Later we introduce existential quantification ("there exists", df-ex 1813) 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 1813. 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 2222) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as Theorem spw 2067 below). Theorem spw 2067 can be used to prove any instance of sp 2222 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2222 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2222 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 2222 as Theorem axc5 39699 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 1828, ax-4 1842, ax-5 1943, ax-6 2000, ax-7 2041, ax-8 2148, and ax-9 2156. 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 2041, ax-8 2148, and ax-9 2156 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 2222, even though (using spw 2067) we can derive all instances of it that do not 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" disjoint variable condition.) Later we will introduce auxiliary axiom schemes ax-10 2179, ax-11 2195, ax-12 2216, and ax-13 2407 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 1568 for use by df-tru 1573. See the comments in that section. In this section, we continue with the first "real" use of it.
1.4.1.1  Existential quantifier
Syntax	wex 1812	Extend wff definition to include the existential quantifier ("there exists").
wff ∃𝑥𝜑
Definition	df-ex 1813	Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". Dual of alex 1859. See also the dual pair alnex 1814 / exnal 1860. Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
Theorem	alnex 1814	Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal 1860 (but does not depend on ax-4 1842 contrary to it). See also the dual pair df-ex 1813 / alex 1859. Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
Theorem	eximal 1815	An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of nonfreeness. See also alimex 1864. (Contributed by BJ, 12-May-2019.)
⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
1.4.1.2  Nonfreeness predicate
Syntax	wnf 1816	Extend wff definition to include the not-free predicate.
wff Ⅎ𝑥𝜑
Definition	df-nf 1817	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 2309). An example of where this is used is stdpc5 2247. See nf5 2320 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 considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered free in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 2046. This definition of "not free" tightly ties to the quantifier ∀𝑥. At this state (no axioms restricting quantifiers yet) "nonfree" 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 2009. This predicate only applies to wffs. See df-nfc 2915 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.)
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	nf2 1818	Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
Theorem	nf3 1819	Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Theorem	nf4 1820	Alternate definition of nonfreeness. This definition uses only primitive symbols (→ , ¬ , ∀). (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	nfi 1821	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfri 1822	Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	nfd 1823	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfrd 1824	Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	nftht 1825	Closed form of nfth 1834. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 3-Sep-2022.)
⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht 1826	Closed form of nfnth 1835. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht2 1827	Closed form of nfnth 1835. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)
⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
1.4.2  Rule scheme ax-gen (Generalization)
Axiom	ax-gen 1828	Rule of (universal) generalization. In our axiomatization, this is the only postulated (that is, axiomatic) rule of inference of predicate calculus (together with the rule of modus ponens ax-mp 5 of propositional 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, then we can conclude ∀𝑥𝑥 = 𝑥 or even ∀𝑦𝑥 = 𝑥. Theorem altru 1840 shows the special case ∀𝑥⊤. The converse rule of inference spi 2223 (universal instantiation, or universal specialization) shows that we can also go the other way: in other words, we can add or remove universal quantifiers from the beginning of any theorem as required. Note that the closed form (𝜑 → ∀𝑥𝜑) need not hold (but may hold in special cases, see ax-5 1943). (Contributed by NM, 3-Jan-1993.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥𝜑
Theorem	gen2 1829	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥∀𝑦𝜑
Theorem	mpg 1830	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
⊢ (∀𝑥𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	mpgbi 1831	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
⊢ (∀𝑥𝜑 ↔ 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	mpgbir 1832	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
Theorem	nex 1833	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ ∃𝑥𝜑
Theorem	nfth 1834	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfnth 1835	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	hbth 1836	No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting "hb...", allow 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.) This hb* idiom is generally being replaced by the nf* idiom (see nfth 1834), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.)
⊢ 𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nftru 1837	The true constant has no free variables. (This can also be proven in one step with nfv 1947, but this proof does not use ax-5 1943.) (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑥⊤
Theorem	nffal 1838	The false constant has no free variables (see nftru 1837). (Contributed by BJ, 6-May-2019.)
⊢ Ⅎ𝑥⊥
Theorem	sptruw 1839	Version of sp 2222 when 𝜑 is true. Instance of a1i 11. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
⊢ 𝜑    ⇒   ⊢ (∀𝑥𝜑 → 𝜑)
Theorem	altru 1840	For all sets, ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ∀𝑥⊤
Theorem	alfal 1841	For all sets, ¬ ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
⊢ ∀𝑥 ¬ ⊥
1.4.3  Axiom scheme ax-4 (Quantified Implication)
Axiom	ax-4 1842	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1843 for labeling consistency. It should be used only by alim 1843. (Contributed by NM, 21-May-2008.) Use alim 1843 instead. (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	alim 1843	Restatement of Axiom ax-4 1842, for labeling consistency. It should be the only theorem using ax-4 1842. (Contributed by NM, 10-Jan-1993.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	alimi 1844	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	2alimi 1845	Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	ala1 1846	Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))
Theorem	al2im 1847	Closed form of al2imi 1848. Version of alim 1843 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
Theorem	al2imi 1848	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alanimi 1849	Variant of al2imi 1848 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
Theorem	alimdh 1850	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1843. (Contributed by NM, 4-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	albi 1851	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
Theorem	albii 1852	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
Theorem	2albii 1853	Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)
Theorem	3albii 1854	Inference adding three universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 10-Aug-2018.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑥∀𝑦∀𝑧𝜓)
Theorem	sylgt 1855	Closed form of sylg 1856. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜓 → 𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
Theorem	sylg 1856	A syllogism combined with generalization. Inference associated with sylgt 1855. General form of alrimih 1857. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1857. (Revised by BJ, 4-Oct-2019.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → ∀𝑥𝜒)
Theorem	alrimih 1857	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2246 and 19.21h 2325. Instance of sylg 1856. (Contributed by NM, 9-Jan-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	hbxfrbi 1858	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2896 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	alex 1859	Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1813. See also the dual pair alnex 1814 / exnal 1860. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Theorem	exnal 1860	Existential quantification of negation is equivalent to negation of universal quantification. Dual of alnex 1814. See also the dual pair df-ex 1813 / alex 1859. Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
Theorem	2nalexn 1861	Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑)
Theorem	2exnaln 1862	Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
Theorem	2nexaln 1863	Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
Theorem	alimex 1864	An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of nonfreeness. See also eximal 1815. (Contributed by BJ, 12-May-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
Theorem	aleximi 1865	A variant of al2imi 1848: instead of applying ∀𝑥 quantifiers to the final implication, replace them with ∃𝑥. A shorter proof is possible using nfa1 2189, sps 2224 and eximd 2255, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	alexbii 1866	Biconditional form of aleximi 1865. (Contributed by BJ, 16-Nov-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exim 1867	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	eximi 1868	Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
Theorem	2eximi 1869	Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)
Theorem	eximii 1870	Inference associated with eximi 1868. (Contributed by BJ, 3-Feb-2018.)
⊢ ∃𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ ∃𝑥𝜓
Theorem	exa1 1871	Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
⊢ (∃𝑥𝜑 → ∃𝑥(𝜓 → 𝜑))
Theorem	19.38 1872	Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1873 and 19.38b 1874. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2245. (Revised by Wolf Lammen, 2-Jan-2018.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	19.38a 1873	Under a nonfreeness hypothesis, the implication 19.38 1872 can be strengthened to an equivalence. See also 19.38b 1874. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	19.38b 1874	Under a nonfreeness hypothesis, the implication 19.38 1872 can be strengthened to an equivalence. See also 19.38a 1873. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	imnang 1875	Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
Theorem	alinexa 1876	A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
Theorem	exnalimn 1877	Existential quantification of a conjunction expressed with only primitive symbols (→, ¬, ∀). (Contributed by NM, 10-May-1993.) State the most general instance. (Revised by BJ, 29-Sep-2019.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))
Theorem	alexn 1878	A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
Theorem	2exnexn 1879	Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
⊢ (∃𝑥∀𝑦𝜑 ↔ ¬ ∀𝑥∃𝑦 ¬ 𝜑)
Theorem	exbi 1880	Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
Theorem	exbii 1881	Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)
Theorem	2exbii 1882	Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)
Theorem	3exbii 1883	Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)
Theorem	nfbiit 1884	Equivalence theorem for the nonfreeness predicate. Closed form of nfbii 1885. (Contributed by Giovanni Mascellani, 10-Apr-2018.) Reduce axiom usage. (Revised by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
Theorem	nfbii 1885	Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
Theorem	nfxfr 1886	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfxfrd 1887	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜒 → Ⅎ𝑥𝜑)
Theorem	nfnbi 1888	A variable is nonfree in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 6-Oct-2024.)
⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
Theorem	nfnt 1889	If a variable is nonfree in a proposition, then it is nonfree in its negation. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1817 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
Theorem	nfn 1890	Inference associated with nfnt 1889. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 ¬ 𝜑
Theorem	nfnd 1891	Deduction associated with nfnt 1889. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
Theorem	exanali 1892	A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))
Theorem	2exanali 1893	Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
Theorem	exancom 1894	Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
Theorem	exan 1895	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 6-Nov-2022.) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023.)
⊢ ∃𝑥𝜑    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	alrimdh 1896	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2246 and 19.21h 2325. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximdh 1897	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	nexdh 1898	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	albidh 1899	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidh 1900	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))