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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 1734) 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 1702) 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 2051) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1964 below).

Theorem spw 1964 can be used to prove any instance of sp 2051 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2051 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2051 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 2051 as theorem axc5 33697 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 1719, ax-4 1734, ax-5 1836, ax-6 1885, ax-7 1932, ax-8 1989, and ax-9 1996. 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 1932, ax-8 1989, and ax-9 1996 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 2051, even though (using spw 1964) 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 2016, ax-11 2031, ax-12 2044, and ax-13 2245 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 1478 for use by df-tru 1483. 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 1701 Extend wff definition to include the existential quantifier ("there exists").
wff 𝑥𝜑

Definitiondf-ex 1702 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 1703 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)

Theoremeximal 1704 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 1755. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

1.4.1.2  Non-freeness predicate

Syntaxwnf 1705 Extend wff definition to include the not-free predicate.
wff 𝑥𝜑

SyntaxwnfOLD 1706 Extend wff definition to include the old not-free predicate. Obsolete as of 16-Sep-2021. (New usage is discouraged.)
wff 𝑥𝜑

Definitiondf-nf 1707 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 2379). An example of where this is used is stdpc5 2074. See nf5 2113 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 1937), 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 1889.

This predicate only applies to wffs. See df-nfc 2750 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 1708 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Theoremnf3 1709 Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Theoremnf4 1710 Alternate definition of non-freeness. This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Theoremnfi 1711 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremnfri 1712 Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
𝑥𝜑       (∃𝑥𝜑 → ∀𝑥𝜑)

Theoremnfd 1713 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
(𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnfrd 1714 Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Theoremnftht 1715 Closed form of nfth 1724. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.)
(∀𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremnfntht 1716 Closed form of nfnth 1725. (Contributed by BJ, 16-Sep-2021.)
(¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Theoremnfntht2 1717 Closed form of nfnth 1725. (Contributed by BJ, 16-Sep-2021.)
(∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Definitiondf-nfOLD 1718 Obsolete definition replaced by nf5 2113 as of 3-Oct-2021. This definition is less suitable than df-nf 1707 when ax-10 2016 and ax-12 2044 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 1719 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 32095 shows the special case 𝑥. Theorem spi 2052 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 1720 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
𝜑       𝑥𝑦𝜑

Theoremmpg 1721 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓

Theoremmpgbi 1722 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 1723 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 1724 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
𝜑       𝑥𝜑

Theoremnfnth 1725 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
¬ 𝜑       𝑥𝜑

Theoremhbth 1726 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 1727 The true constant has no free variables. (This can also be proven in one step with nfv 1840, but this proof does not use ax-5 1836.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥

Theoremnex 1728 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑

Theoremnffal 1729 The false constant has no free variables (see nftru 1727). (Contributed by BJ, 6-May-2019.)
𝑥

Theoremsptruw 1730 Version of sp 2051 when 𝜑 is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
𝜑       (∀𝑥𝜑𝜑)

TheoremnfiOLD 1731 Obsolete proof of nf5i 2021 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑

TheoremnfthOLD 1732 Obsolete proof of nfth 1724 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝑥𝜑

TheoremnfnthOLD 1733 Obsolete proof of nfnth 1725 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 1734 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1735 for labeling consistency. It should be used only by alim 1735. (Contributed by NM, 21-May-2008.) Use alim 1735 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalim 1735 Restatement of Axiom ax-4 1734, for labeling consistency. It should be the only theorem using ax-4 1734. (Contributed by NM, 10-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalimi 1736 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theorem2alimi 1737 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)

Theoremala1 1738 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥𝜑 → ∀𝑥(𝜓𝜑))

Theoremal2im 1739 Closed form of al2imi 1740. Version of alim 1735 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Theoremal2imi 1740 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalanimi 1741 Variant of al2imi 1740 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Theoremalimdh 1742 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1735. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalbi 1743 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Theoremalbii 1744 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Theorem2albii 1745 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓)

Theoremsylgt 1746 Closed form of sylg 1747. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))

Theoremsylg 1747 A syllogism combined with generalization. Inference associated with sylgt 1746. General form of alrimih 1748. (Contributed by BJ, 4-Oct-2019.)
(𝜑 → ∀𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝜒)

Theoremalrimih 1748 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2073 and 19.21h 2118. Instance of sylg 1747. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremhbxfrbi 1749 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2727 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑 → ∀𝑥𝜑)

Theoremalex 1750 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 1751 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)

Theorem2nalexn 1752 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)

Theorem2exnaln 1753 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Theorem2nexaln 1754 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremalimex 1755 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 1704. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Theoremaleximi 1756 A variant of al2imi 1740: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2025, sps 2053 and eximd 2083, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremalexbii 1757 Biconditional form of aleximi 1756. (Contributed by BJ, 16-Nov-2020.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexim 1758 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Theoremeximi 1759 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑𝜓)       (∃𝑥𝜑 → ∃𝑥𝜓)

Theorem2eximi 1760 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓)

Theoremeximii 1761 Inference associated with eximi 1759. (Contributed by BJ, 3-Feb-2018.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓

Theoremexa1 1762 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∃𝑥𝜑 → ∃𝑥(𝜓𝜑))

Theorem19.38 1763 Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1764 and 19.38b 1765. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2071. (Revised by Wolf Lammen, 2-Jan-2018.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.38a 1764 Under a non-freeness hypothesis, the implication 19.38 1763 can be strengthened to an equivalence. See also 19.38b 1765. (Contributed by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theorem19.38b 1765 Under a non-freeness hypothesis, the implication 19.38 1763 can be strengthened to an equivalence. See also 19.38a 1764. (Contributed by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theoremimnang 1766 Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))

Theoremalinexa 1767 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))

Theoremalexn 1768 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
(∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)

Theorem2exnexn 1769 Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Theoremexbi 1770 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Theoremexbii 1771 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(𝜑𝜓)       (∃𝑥𝜑 ↔ ∃𝑥𝜓)

Theorem2exbii 1772 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓)

Theorem3exbii 1773 Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)

Theoremnfbiit 1774 Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1775. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremnfbii 1775 Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Theoremnfxfr 1776 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑

Theoremnfxfrd 1777 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑𝜓)    &   (𝜒 → Ⅎ𝑥𝜓)       (𝜒 → Ⅎ𝑥𝜑)

Theoremnfnbi 1778 A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Theoremnfnt 1779 If a variable is non-free in a proposition, then it is non-free 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 1707 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

TheoremnfntOLDOLD 1780 Obsolete proof of nfnt 1779 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1707 changed. (Revised by Wolf Lammen, 4-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Theoremnfn 1781 Inference associated with nfnt 1779. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       𝑥 ¬ 𝜑

Theoremnfnd 1782 Deduction associated with nfnt 1779. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremexanali 1783 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(𝜑𝜓))

Theoremexancom 1784 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))

Theoremexan 1785 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, 8-Oct-2021.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

TheoremexanOLD 1786 Obsolete proof of exan 1785 as of 8-Oct-2021. (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 modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

Theoremalrimdh 1787 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073 and 19.21h 2118. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximdh 1788 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremnexdh 1789 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbidh 1790 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidh 1791 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexsimpl 1792 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜑)

Theoremexsimpr 1793 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜓)

Theorem19.40 1794 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Theorem19.26 1795 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Theorem19.26-2 1796 Theorem 19.26 1795 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))

Theorem19.26-3an 1797 Theorem 19.26 1795 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theorem19.29 1798 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1799. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r 1799 Variation of 19.29 1798. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r2 1800 Variation of 19.29r 1799 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

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