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

Theoremsbid 1701 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremstdpc4 1702 The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑦), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥)." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Theoremsbh 1703 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥]𝜑𝜑)

Theoremsbf 1704 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 1705 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Theoremsb6x 1706 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 1707 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 1708 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 1709 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 1710 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 1711 A substitution into a theorem remains true. (See chvar 1684 and chvarv 1857 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
𝜑       [𝑦 / 𝑥]𝜑

Theoremequsb1 1712 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[𝑦 / 𝑥]𝑥 = 𝑦

Theoremequsb2 1713 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
[𝑦 / 𝑥]𝑦 = 𝑥

Theoremsbiedh 1714 Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1717). New proofs should use sbied 1715 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 1715 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1718). (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 1716* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1718). (Contributed by NM, 7-Jan-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbieh 1717 Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1718 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbie 1718 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 1719 A property related to substitution that unlike equs5 1754 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5e 1720 A property related to substitution that unlike equs5 1754 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 1721 Analogue to ax-11 1440 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 1722 Quantifier Substitution for existential quantifiers. Analogue to ax10o 1647 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))

Theoremdrex1 1723 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 1724 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 1725 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 1726 A version of sb4 1757 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs45f 1727 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 1728 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 1729 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 1730 One direction of a simplified definition of substitution that unlike sb4 1757 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremhbsb2a 1731 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremhbsb2e 1732 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Theoremhbsb3 1733 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 1734 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 1735 Version of sbco 1887 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 1736* A version of spim 1670 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremaev 1737* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1739. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Theoremax16 1738* Theorem showing that ax-16 1739 is redundant if ax-17 1462 is included in the axiom system. The important part of the proof is provided by aev 1737.

See ax16ALT 1784 for an alternate proof that does not require ax-10 1439 or ax-12 1445.

This theorem should not be referenced in any proof. Instead, use ax-16 1739 below so that theorems needing ax-16 1739 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Axiomax-16 1739* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1462 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 1462; see theorem ax16 1738.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1738. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremdveeq2 1740* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremdveeq2or 1741* Quantifier introduction when one pair of variables is distinct. Like dveeq2 1740 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 1742* Proof of dvelimf 1936 using dveeq2 1740 (shown as the last hypothesis) instead of ax-12 1445. This shows that ax-12 1445 could be replaced by dveeq2 1740 (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 1743* A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
(¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremexlimdv 1744* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremax11v2 1745* Recovery of ax11o 1747 from ax11v 1752 without using ax-11 1440. The hypothesis is even weaker than ax11v 1752, 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 1747. (Contributed by NM, 2-Feb-2007.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax11a2 1746* Derive ax-11o 1748 from a hypothesis in the form of ax-11 1440. The hypothesis is even weaker than ax-11 1440, 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 1747. (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 1747 Derivation of set.mm's original ax-11o 1748 from the shorter ax-11 1440 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1739 or ax-17 1462.

Normally, ax11o 1747 should be used rather than ax-11o 1748, 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 1748 Axiom ax-11o 1748 ("o" for "old") was the original version of ax-11 1440, before it was discovered (in Jan. 2007) that the shorter ax-11 1440 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 1747.

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1747. (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 1749* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidv 1750* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremax11b 1751 A bidirectional version of ax-11o 1748. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax11v 1752* This is a version of ax-11o 1748 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 1753* Analogue to ax11v 1752 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
(𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))

Theoremequs5 1754 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremequs5or 1755 Lemma used in proofs of substitution properties. Like equs5 1754 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 1756 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 1757 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 1758 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1757 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
(∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4b 1759 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4bor 1760 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 1761 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theoremnfsb2or 1762 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1761 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 1763 Propositional logic lemma used in the sbequi 1764 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
(𝜑 ∨ (𝜓 → (𝜒𝜃)))    &   (𝜏 ∨ (𝜓 → (𝜃𝜂)))       (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒𝜂))))

Theoremsbequi 1764 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 1765 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 1766 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 1767 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 1768 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 1769 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 1770 Deduction substituting both sides of a biconditional. New proofs should use sbbid 1771 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbbid 1771 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbequ8 1772 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 1773 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 1774 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbid2 1775 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 1776 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 1777 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 1778 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 1779 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 1780 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 1781 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 1782 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 1783* Inference with ax-16 1739 as its conclusion, that doesn't require ax-10 1439, ax-11 1440, or ax-12 1445 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 1784* Version of ax16 1738 that doesn't require ax-10 1439 or ax-12 1445 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremspv 1785* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 1786* Distinct-variable version of spime 1673. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspeiv 1787* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremequvin 1788* 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 1789* A generalization of axiom ax-16 1739. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theorema16gb 1790* A generalization of axiom ax-16 1739. (Contributed by NM, 5-Aug-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Theorema16nf 1791* If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theorem2albidv 1792* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))

Theorem2exbidv 1793* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))

Theorem3exbidv 1794* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))

Theorem4exbidv 1795* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))

Theorem19.9v 1796* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
(∃𝑥𝜑𝜑)

Theoremexlimdd 1797 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theorem19.21v 1798* 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 1518 via the use of distinct variable conditions combined with ax-17 1462. 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 1950 derived from df-eu 1948. 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 1799* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimivv 1800* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)

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