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Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremax11v2 1701* Recovery of ax11o 1703 from ax11v 1708 without using ax-11 1397. The hypothesis is even weaker than ax11v 1708, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1703. (Contributed by NM, 2-Feb-2007.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax11a2 1702* Derive ax-11o 1704 from a hypothesis in the form of ax-11 1397. The hypothesis is even weaker than ax-11 1397, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1703. (Contributed by NM, 2-Feb-2007.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

1.4.2  Derive the obsolete axiom of variable substitution ax-11o

Theoremax11o 1703 Derivation of set.mm's original ax-11o 1704 from the shorter ax-11 1397 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1695 or ax-17 1419.

Normally, ax11o 1703 should be used rather than ax-11o 1704, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Axiomax-11o 1704 Axiom ax-11o 1704 ("o" for "old") was the original version of ax-11 1397, before it was discovered (in Jan. 2007) that the shorter ax-11 1397 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1703.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1703. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

1.4.3  More theorems related to ax-11 and substitution

Theoremalbidv 1705* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidv 1706* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremax11b 1707 A bidirectional version of ax-11o 1704. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax11v 1708* This is a version of ax-11o 1704 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax11ev 1709* Analogue to ax11v 1708 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
(𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))

Theoremequs5 1710 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremequs5or 1711 Lemma used in proofs of substitution properties. Like equs5 1710 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb3 1712 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Theoremsb4 1713 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4or 1714 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1713 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4b 1715 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4bor 1716 Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremhbsb2 1717 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theoremnfsb2or 1718 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1717 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremsbequilem 1719 Propositional logic lemma used in the sbequi 1720 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
(𝜑 ∨ (𝜓 → (𝜒𝜃)))    &   (𝜏 ∨ (𝜓 → (𝜃𝜂)))       (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒𝜂))))

Theoremsbequi 1720 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Theoremsbequ 1721 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremdrsb2 1722 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremspsbe 1723 A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremspsbim 1724 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremspsbbi 1725 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Theoremsbbid 1726 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbequ8 1727 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))

Theoremsbft 1728 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))

Theoremsbid2h 1729 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbid2 1730 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbidm 1731 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsb5rf 1732 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → ∀𝑦𝜑)       (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Theoremsb6rf 1733 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → ∀𝑦𝜑)       (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Theoremsb8h 1734 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
(𝜑 → ∀𝑦𝜑)       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8eh 1735 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
(𝜑 → ∀𝑦𝜑)       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8 1736 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8e 1737 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

1.4.4  Predicate calculus with distinct variables (cont.)

Theoremax16i 1738* Inference with ax-16 1695 as its conclusion, that doesn't require ax-10 1396, ax-11 1397, or ax-12 1402 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremax16ALT 1739* Version of ax16 1694 that doesn't require ax-10 1396 or ax-12 1402 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremspv 1740* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 1741* Distinct-variable version of spime 1629. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspeiv 1742* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremequvin 1743* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Theorema16g 1744* A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorema16gb 1745* A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Theorema16nf 1746* If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theorem2albidv 1747* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))

Theorem2exbidv 1748* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))

Theorem3exbidv 1749* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))

Theorem4exbidv 1750* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))

Theorem19.9v 1751* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
(∃𝑥𝜑𝜑)

Theoremexlimdd 1752 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theorem19.21v 1753* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as (𝜑 → ∀𝑥𝜑) in 19.21 1475 via the use of distinct variable conditions combined with ax-17 1419. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 1905 derived from df-eu 1903. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremalrimiv 1754* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimivv 1755* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)

Theoremalrimdv 1756* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremnfdv 1757* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theorem2ax17 1758* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)

Theoremalimdv 1759* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremeximdv 1760* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theorem2alimdv 1761* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))

Theorem2eximdv 1762* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))

Theorem19.23v 1763* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.23vv 1764* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theoremsb56 1765* Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1646. (Contributed by NM, 14-Apr-2008.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb6 1766* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5 1767* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsbnv 1768* Version of sbn 1826 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Theoremsbanv 1769* Version of sban 1829 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsborv 1770* Version of sbor 1828 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Theoremsbi1v 1771* Forward direction of sbimv 1773. (Contributed by Jim Kingdon, 25-Dec-2017.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbi2v 1772* Reverse direction of sbimv 1773. (Contributed by Jim Kingdon, 18-Jan-2018.)
(([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))

Theoremsbimv 1773* Intuitionistic proof of sbim 1827 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsblimv 1774* Version of sblim 1831 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
(𝜓 → ∀𝑥𝜓)       ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))

Theorempm11.53 1775* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theoremexlimivv 1776* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)

Theoremexlimdvv 1777* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))

Theoremexlimddv 1778* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theorem19.27v 1779* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28v 1780* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.36aiv 1781* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.41v 1782* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41vv 1783* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.41vvv 1784* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))

Theorem19.41vvvv 1785* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))

Theorem19.42v 1786* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexdistr 1787* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theorem19.42vv 1788* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))

Theorem19.42vvv 1789* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))

Theorem19.42vvvv 1790* Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))

Theoremexdistr2 1791* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))

Theorem3exdistr 1792* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Theorem4exdistr 1793* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

Theoremcbvalv 1794* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv 1795* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbval2 1796* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2 1797* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbval2v 1798* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2v 1799* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvald 1800* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1893. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

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