P. 18 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	sbid 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]𝜑 ↔ 𝜑)
Theorem	stdpc4 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)
Theorem	sbh 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]𝜑 ↔ 𝜑)
Theorem	sbf 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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbf2 1705	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	sb6x 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 = 𝑦 → 𝜑))
Theorem	nfs1f 1707	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 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)
Theorem	sbequ5 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)
Theorem	sbequ6 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)
Theorem	sbt 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
Theorem	equsb1 1712	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
⊢ [𝑦 / 𝑥]𝑥 = 𝑦
Theorem	equsb2 1713	Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)
⊢ [𝑦 / 𝑥]𝑦 = 𝑥
Theorem	sbiedh 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 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbied 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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbiedv 1716*	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1718). (Contributed by NM, 7-Jan-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbieh 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbie 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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
1.3.11  Theorems using axiom ax-11
Theorem	equs5a 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 = 𝑦 → 𝜑))
Theorem	equs5e 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 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	ax11e 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 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑))
Theorem	ax10oe 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))
Theorem	drex1 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drsb1 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))
Theorem	exdistrfor 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 ∨ ∀𝑥Ⅎ𝑦𝜑)    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	sb4a 1726	A version of sb4 1757 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs45f 1727	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 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 = 𝑦 → 𝜑))
Theorem	sb5f 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 = 𝑦 ∧ 𝜑))
Theorem	sb4e 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))
Theorem	hbsb2a 1731	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 1732	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 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)
Theorem	nfs1 1734	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 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
Theorem	spimv 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 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	aev 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)
Theorem	ax16 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))
Axiom	ax-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))
Theorem	dveeq2 1740*	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 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)
Theorem	dvelimfALT2 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 → (𝜑 ↔ 𝜓))    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	nd5 1743*	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	exlimdv 1744*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	ax11v2 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 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax11a2 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 = 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 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 = 𝑦 → 𝜑))))
Axiom	ax-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 = 𝑦 → 𝜑))))
1.4.3  More theorems related to ax-11 and substitution
Theorem	albidv 1749*	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1750*	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	ax11b 1751	A bidirectional version of ax-11o 1748. (Contributed by NM, 30-Jun-2006.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax11v 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 = 𝑦 → 𝜑)))
Theorem	ax11ev 1753*	Analogue to ax11v 1752 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
Theorem	equs5 1754	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 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 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	sb3 1756	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 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 = 𝑦 → 𝜑)))
Theorem	sb4or 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 = 𝑦 → 𝜑)))
Theorem	sb4b 1759	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 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 = 𝑦 → 𝜑)))
Theorem	hbsb2 1761	Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	nfsb2or 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)
Theorem	sbequilem 1763	Propositional logic lemma used in the sbequi 1764 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
⊢ (𝜑 ∨ (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜏 ∨ (𝜓 → (𝜃 → 𝜂)))    ⇒   ⊢ (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒 → 𝜂))))
Theorem	sbequi 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))
Theorem	sbequ 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))
Theorem	drsb2 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))
Theorem	spsbe 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)
Theorem	spsbim 1768	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 1769	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 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 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	sbbid 1771	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 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 = 𝑦 → 𝜑))
Theorem	sbft 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]𝜑 ↔ 𝜑))
Theorem	sbid2h 1774	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)
Theorem	sbid2 1775	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 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)
Theorem	sb5rf 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))
Theorem	sb6rf 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))
Theorem	sb8h 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)
Theorem	sb8eh 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)
Theorem	sb8 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.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8e 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.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
1.4.4  Predicate calculus with distinct variables (cont.)
Theorem	ax16i 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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	ax16ALT 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))
Theorem	spv 1785*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimev 1786*	Distinct-variable version of spime 1673. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	speiv 1787*	Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	equvin 1788*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
Theorem	a16g 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))
Theorem	a16gb 1790*	A generalization of axiom ax-16 1739. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Theorem	a16nf 1791*	If there is only one element in the universe, then everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Theorem	2albidv 1792*	Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 ↔ ∀𝑥∀𝑦𝜒))
Theorem	2exbidv 1793*	Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))
Theorem	3exbidv 1794*	Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))
Theorem	4exbidv 1795*	Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))
Theorem	19.9v 1796*	Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	exlimdd 1797	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	19.21v 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(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	alrimiv 1799*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1800*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)