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

Theoremsbf 1701 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑𝜑)

Theoremsbf2 1702 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Theoremsb6x 1703 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremnfs1f 1704 If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥[𝑦 / 𝑥]𝜑

Theoremhbs1f 1705 If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremsbequ5 1706 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.)
([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremsbequ6 1707 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.)
([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremsbt 1708 A substitution into a theorem remains true. (See chvar 1681 and chvarv 1854 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝜑       [𝑦 / 𝑥]𝜑

Theoremequsb1 1709 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[𝑦 / 𝑥]𝑥 = 𝑦

Theoremequsb2 1710 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[𝑦 / 𝑥]𝑦 = 𝑥

Theoremsbiedh 1711 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1714). New proofs should use sbied 1712 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbied 1712 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1715). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbiedv 1713* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1715). (Contributed by NM, 7-Jan-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbieh 1714 Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1715 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbie 1715 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

1.3.11  Theorems using axiom ax-11

Theoremequs5a 1716 A property related to substitution that unlike equs5 1751 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5e 1717 A property related to substitution that unlike equs5 1751 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremax11e 1718 Analogue to ax-11 1438 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)
(𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑦𝜑))

Theoremax10oe 1719 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1644 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))

Theoremdrex1 1720 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.) (Revised by NM, 3-Feb-2015.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theoremdrsb1 1721 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, 5-Aug-1993.)
(∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Theoremexdistrfor 1722 Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Jim Kingdon, 25-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦𝜑)       (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theoremsb4a 1723 A version of sb4 1754 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs45f 1724 Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.)
(𝜑 → ∀𝑦𝜑)       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb6f 1725 Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5f 1726 Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsb4e 1727 One direction of a simplified definition of substitution that unlike sb4 1754 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremhbsb2a 1728 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremhbsb2e 1729 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Theoremhbsb3 1730 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1 1731 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑

Theoremsbcof2 1732 Version of sbco 1884 where 𝑥 is not free in 𝜑. (Contributed by Jim Kingdon, 28-Dec-2017.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

1.4  Predicate calculus with distinct variables

1.4.1  Derive the axiom of distinct variables ax-16

Theoremspimv 1733* A version of spim 1667 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremaev 1734* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1736. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Theoremax16 1735* Theorem showing that ax-16 1736 is redundant if ax-17 1460 is included in the axiom system. The important part of the proof is provided by aev 1734.

See ax16ALT 1781 for an alternate proof that does not require ax-10 1437 or ax-12 1443.

This theorem should not be referenced in any proof. Instead, use ax-16 1736 below so that theorems needing ax-16 1736 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Axiomax-16 1736* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1460 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory, but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1460; see theorem ax16 1735.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1735. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremdveeq2 1737* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremdveeq2or 1738* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1737 but connecting 𝑥𝑥 = 𝑦 by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)

TheoremdvelimfALT2 1739* Proof of dvelimf 1933 using dveeq2 1737 (shown as the last hypothesis) instead of ax-12 1443. This shows that ax-12 1443 could be replaced by dveeq2 1737 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremnd5 1740* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
(¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremexlimdv 1741* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremax11v2 1742* Recovery of ax11o 1744 from ax11v 1749 without using ax-11 1438. The hypothesis is even weaker than ax11v 1749, 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 1744. (Contributed by NM, 2-Feb-2007.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax11a2 1743* Derive ax-11o 1745 from a hypothesis in the form of ax-11 1438. The hypothesis is even weaker than ax-11 1438, 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 1744. (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 1744 Derivation of set.mm's original ax-11o 1745 from the shorter ax-11 1438 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1736 or ax-17 1460.

Normally, ax11o 1744 should be used rather than ax-11o 1745, 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 1745 Axiom ax-11o 1745 ("o" for "old") was the original version of ax-11 1438, before it was discovered (in Jan. 2007) that the shorter ax-11 1438 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 1744.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1744. (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 1746* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidv 1747* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremax11b 1748 A bidirectional version of ax-11o 1745. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax11v 1749* This is a version of ax-11o 1745 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 1750* Analogue to ax11v 1749 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
(𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))

Theoremequs5 1751 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremequs5or 1752 Lemma used in proofs of substitution properties. Like equs5 1751 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 1753 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 1754 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 1755 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1754 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4b 1756 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4bor 1757 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 1758 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theoremnfsb2or 1759 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1758 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 1760 Propositional logic lemma used in the sbequi 1761 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
(𝜑 ∨ (𝜓 → (𝜒𝜃)))    &   (𝜏 ∨ (𝜓 → (𝜃𝜂)))       (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒𝜂))))

Theoremsbequi 1761 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 1762 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 1763 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 1764 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 1765 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 1766 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))

Theoremsbbidh 1767 Deduction substituting both sides of a biconditional. New proofs should use sbbid 1768 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbbid 1768 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbequ8 1769 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 1770 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 1771 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbid2 1772 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 1773 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 1774 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 1775 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 1776 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 1777 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 1778 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 1779 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 1780* Inference with ax-16 1736 as its conclusion, that doesn't require ax-10 1437, ax-11 1438, or ax-12 1443 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 1781* Version of ax16 1735 that doesn't require ax-10 1437 or ax-12 1443 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremspv 1782* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 1783* Distinct-variable version of spime 1670. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspeiv 1784* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremequvin 1785* 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 1786* A generalization of axiom ax-16 1736. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorema16gb 1787* A generalization of axiom ax-16 1736. (Contributed by NM, 5-Aug-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Theorema16nf 1788* If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theorem2albidv 1789* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))

Theorem2exbidv 1790* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))

Theorem3exbidv 1791* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))

Theorem4exbidv 1792* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))

Theorem19.9v 1793* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
(∃𝑥𝜑𝜑)

Theoremexlimdd 1794 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theorem19.21v 1795* 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 1516 via the use of distinct variable conditions combined with ax-17 1460. 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 1947 derived from df-eu 1945. 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 1796* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimivv 1797* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)

Theoremalrimdv 1798* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremnfdv 1799* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theorem2ax17 1800* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)

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