P. 19 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1801-1900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbval 1801	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvexh 1802	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbvex 1803	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	chvar 1804	Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	equvini 1805	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦 (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
Theorem	equveli 1806	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1805.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
⊢ (∀𝑧(𝑧 = 𝑥 ↔ 𝑧 = 𝑦) → 𝑥 = 𝑦)
Theorem	nfald 1807	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	nfexd 1808	If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
1.3.10  Substitution (without distinct variables)
Syntax	wsb 1809	Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.)
wff [𝑦 / 𝑥]𝜑
Definition	df-sb 1810	Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff that results when 𝑦 is properly substituted for 𝑥 in the wff 𝜑". We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1822. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see Theorems sbequ 1887, sbcom2 2039 and sbid2v 2048). Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1821 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2043 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 2046. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1935 and sb6 1934. In classical logic, another possible definition is (𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 5-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	sbimi 1811	Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
Theorem	sbbii 1812	Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
Theorem	sb1 1813	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	sb2 1814	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
Theorem	sbequ1 1815	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
Theorem	sbequ2 1816	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
Theorem	stdpc7 1817	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1750.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑦)". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))
Theorem	sbequ12 1818	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
Theorem	sbequ12r 1819	An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
Theorem	sbequ12a 1820	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
Theorem	sbid 1821	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]𝜑 ↔ 𝜑)
Theorem	stdpc4 1822	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)
Theorem	sbh 1823	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]𝜑 ↔ 𝜑)
Theorem	sbf 1824	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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbf2 1825	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	sb6x 1826	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 = 𝑦 → 𝜑))
Theorem	nfs1f 1827	If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	hbs1f 1828	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)
Theorem	sbequ5 1829	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)
Theorem	sbequ6 1830	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)
Theorem	sbt 1831	A substitution into a theorem remains true. (See chvar 1804 and chvarv 1989 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝜑    ⇒   ⊢ [𝑦 / 𝑥]𝜑
Theorem	equsb1 1832	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
⊢ [𝑦 / 𝑥]𝑥 = 𝑦
Theorem	equsb2 1833	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
⊢ [𝑦 / 𝑥]𝑦 = 𝑥
Theorem	sbiedh 1834	Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1837). New proofs should use sbied 1835 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 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbied 1835	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1838). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbiedv 1836*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1838). (Contributed by NM, 7-Jan-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbieh 1837	Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1838 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbie 1838	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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbiev 1839*	Conversion of implicit substitution to explicit substitution. Version of sbie 1838 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	equsalv 1840*	An equivalence related to implicit substitution. Version of equsal 1774 with a disjoint variable condition. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
1.3.11  Theorems using axiom ax-11
Theorem	equs5a 1841	A property related to substitution that unlike equs5 1876 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5e 1842	A property related to substitution that unlike equs5 1876 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	ax11e 1843	Analogue to ax-11 1554 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 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑))
Theorem	ax10oe 1844	Quantifier Substitution for existential quantifiers. Analogue to ax10o 1762 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))
Theorem	drex1 1845	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drsb1 1846	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))
Theorem	exdistrfor 1847	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 ∨ ∀𝑥Ⅎ𝑦𝜑)    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	sb4a 1848	A version of sb4 1879 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs45f 1849	Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	sb6f 1850	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 = 𝑦 → 𝜑))
Theorem	sb5f 1851	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 = 𝑦 ∧ 𝜑))
Theorem	sb4e 1852	One direction of a simplified definition of substitution that unlike sb4 1879 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	hbsb2a 1853	Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	hbsb2e 1854	Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
Theorem	hbsb3 1855	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)
Theorem	nfs1 1856	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	sbcof2 1857	Version of sbco 2020 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
Theorem	spimv 1858*	A version of spim 1785 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	aev 1859*	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1861. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Theorem	ax16 1860*	Theorem showing that ax-16 1861 is redundant if ax-17 1574 is included in the axiom system. The important part of the proof is provided by aev 1859. See ax16ALT 1906 for an alternate proof that does not require ax-10 1553 or ax12 1560. This theorem should not be referenced in any proof. Instead, use ax-16 1861 below so that theorems needing ax-16 1861 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Axiom	ax-16 1861*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1574 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 1574; see Theorem ax16 1860. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1860. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	dveeq2 1862*	Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	dveeq2or 1863*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1862 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)
Theorem	dvelimfALT2 1864*	Proof of dvelimf 2067 using dveeq2 1862 (shown as the last hypothesis) instead of ax12 1560. This shows that ax12 1560 could be replaced by dveeq2 1862 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑧𝜓)    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	nd5 1865*	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	exlimdv 1866*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	ax11v2 1867*	Recovery of ax11o 1869 from ax11v 1874 without using ax-11 1554. The hypothesis is even weaker than ax11v 1874, 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 1869. (Contributed by NM, 2-Feb-2007.)
⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax11a2 1868*	Derive ax-11o 1870 from a hypothesis in the form of ax-11 1554. The hypothesis is even weaker than ax-11 1554, 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 1869. (Contributed by NM, 2-Feb-2007.)
⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
1.4.2  Derive the obsolete axiom of variable substitution ax-11o
Theorem	ax11o 1869	Derivation of set.mm's original ax-11o 1870 from the shorter ax-11 1554 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1861 or ax-17 1574. Normally, ax11o 1869 should be used rather than ax-11o 1870, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Axiom	ax-11o 1870	Axiom ax-11o 1870 ("o" for "old") was the original version of ax-11 1554, before it was discovered (in Jan. 2007) that the shorter ax-11 1554 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 1869. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1869. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
1.4.3  More theorems related to ax-11 and substitution
Theorem	albidv 1871*	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1872*	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	ax11b 1873	A bidirectional version of ax-11o 1870. (Contributed by NM, 30-Jun-2006.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax11v 1874*	This is a version of ax-11o 1870 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 = 𝑦 → 𝜑)))
Theorem	ax11ev 1875*	Analogue to ax11v 1874 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
Theorem	equs5 1876	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	equs5or 1877	Lemma used in proofs of substitution properties. Like equs5 1876 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 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	sb3 1878	One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
Theorem	sb4 1879	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 = 𝑦 → 𝜑)))
Theorem	sb4or 1880	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1879 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	sb4b 1881	Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	sb4bor 1882	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 = 𝑦 → 𝜑)))
Theorem	hbsb2 1883	Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	nfsb2or 1884	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1883 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)
Theorem	sbequilem 1885	Propositional logic lemma used in the sbequi 1886 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
⊢ (𝜑 ∨ (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜏 ∨ (𝜓 → (𝜃 → 𝜂)))    ⇒   ⊢ (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒 → 𝜂))))
Theorem	sbequi 1886	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))
Theorem	sbequ 1887	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))
Theorem	drsb2 1888	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))
Theorem	spsbe 1889	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)
Theorem	spsbim 1890	Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	spsbbi 1891	Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Theorem	sbbidh 1892	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1893 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	sbbid 1893	Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	sbequ8 1894	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 = 𝑦 → 𝜑))
Theorem	sbft 1895	Substitution has no effect on a nonfree 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]𝜑 ↔ 𝜑))
Theorem	sbid2h 1896	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)
Theorem	sbid2 1897	An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)
Theorem	sbidm 1898	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)
Theorem	sb5rf 1899	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))
Theorem	sb6rf 1900	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))