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	ax16 1801*	Theorem showing that ax-16 1802 is redundant if ax-17 1514 is included in the axiom system. The important part of the proof is provided by aev 1800. See ax16ALT 1847 for an alternate proof that does not require ax-10 1493 or ax12 1500. This theorem should not be referenced in any proof. Instead, use ax-16 1802 below so that theorems needing ax-16 1802 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Axiom	ax-16 1802*	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1514 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 1514; see Theorem ax16 1801. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1801. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	dveeq2 1803*	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 1804*	Quantifier introduction when one pair of variables is distinct. Like dveeq2 1803 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 1805*	Proof of dvelimf 2003 using dveeq2 1803 (shown as the last hypothesis) instead of ax12 1500. This shows that ax12 1500 could be replaced by dveeq2 1803 (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 1806*	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	exlimdv 1807*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	ax11v2 1808*	Recovery of ax11o 1810 from ax11v 1815 without using ax-11 1494. The hypothesis is even weaker than ax11v 1815, 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 1810. (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 1809*	Derive ax-11o 1811 from a hypothesis in the form of ax-11 1494. The hypothesis is even weaker than ax-11 1494, 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 1810. (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 1810	Derivation of set.mm's original ax-11o 1811 from the shorter ax-11 1494 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1802 or ax-17 1514. Normally, ax11o 1810 should be used rather than ax-11o 1811, 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 1811	Axiom ax-11o 1811 ("o" for "old") was the original version of ax-11 1494, before it was discovered (in Jan. 2007) that the shorter ax-11 1494 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 1810. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1810. (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 1812*	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidv 1813*	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	ax11b 1814	A bidirectional version of ax-11o 1811. (Contributed by NM, 30-Jun-2006.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax11v 1815*	This is a version of ax-11o 1811 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 1816*	Analogue to ax11v 1815 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
Theorem	equs5 1817	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 1818	Lemma used in proofs of substitution properties. Like equs5 1817 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 1819	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 1820	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 1821	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1820 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 1822	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 1823	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 1824	Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	nfsb2or 1825	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1824 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 1826	Propositional logic lemma used in the sbequi 1827 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
⊢ (𝜑 ∨ (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜏 ∨ (𝜓 → (𝜃 → 𝜂)))    ⇒   ⊢ (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒 → 𝜂))))
Theorem	sbequi 1827	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 1828	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 1829	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 1830	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 1831	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 1832	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 1833	Deduction substituting both sides of a biconditional. New proofs should use sbbid 1834 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 1834	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 1835	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 1836	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 1837	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)
Theorem	sbid2 1838	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 1839	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 1840	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 1841	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 1842	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 1843	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 1844	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 1845	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 1846*	Inference with ax-16 1802 as its conclusion, that does not require ax-10 1493, ax-11 1494, or ax12 1500 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 1847*	Version of ax16 1801 that does not require ax-10 1493 or ax12 1500 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	spv 1848*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimev 1849*	Distinct-variable version of spime 1729. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	speiv 1850*	Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	equvin 1851*	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 1852*	A generalization of Axiom ax-16 1802. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	a16gb 1853*	A generalization of Axiom ax-16 1802. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Theorem	a16nf 1854*	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 1855*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 ↔ ∀𝑥∀𝑦𝜒))
Theorem	2exbidv 1856*	Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))
Theorem	3exbidv 1857*	Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))
Theorem	4exbidv 1858*	Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))
Theorem	19.9v 1859*	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 1860	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 1861*	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 1571 via the use of distinct variable conditions combined with ax-17 1514. 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 2019 derived from df-eu 2017. 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 1862*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimivv 1863*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alrimdv 1864*	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	nfdv 1865*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	2ax17 1866*	Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
Theorem	alimdv 1867*	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	eximdv 1868*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	2alimdv 1869*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	2eximdv 1870*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))
Theorem	19.23v 1871*	Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.23vv 1872*	Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
Theorem	sb56 1873*	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 1751. (Contributed by NM, 14-Apr-2008.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	sb6 1874*	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 = 𝑦 → 𝜑))
Theorem	sb5 1875*	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 = 𝑦 ∧ 𝜑))
Theorem	sbnv 1876*	Version of sbn 1940 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Theorem	sbanv 1877*	Version of sban 1943 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	sborv 1878*	Version of sbor 1942 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
Theorem	sbi1v 1879*	Forward direction of sbimv 1881. (Contributed by Jim Kingdon, 25-Dec-2017.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbi2v 1880*	Reverse direction of sbimv 1881. (Contributed by Jim Kingdon, 18-Jan-2018.)
⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
Theorem	sbimv 1881*	Intuitionistic proof of sbim 1941 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sblimv 1882*	Version of sblim 1945 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	pm11.53 1883*	Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	exlimivv 1884*	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → 𝜓)
Theorem	exlimdvv 1885*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))
Theorem	exlimddv 1886*	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	19.27v 1887*	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28v 1888*	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.36aiv 1889*	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.41v 1890*	Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41vv 1891*	Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))
Theorem	19.41vvv 1892*	Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
Theorem	19.41vvvv 1893*	Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑤∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
Theorem	19.42v 1894*	Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	spvv 1895*	Version of spv 1848 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	chvarvv 1896*	Version of chvarv 1925 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	exdistr 1897*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	exdistrv 1898*	Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1890 and 19.42v 1894. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1920. (Contributed by BJ, 30-Sep-2022.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	19.42vv 1899*	Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))
Theorem	19.42vvv 1900*	Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓))