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	hbnd 1701	Deduction form of bound-variable hypothesis builder hbn 1700. (Contributed by NM, 3-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
Theorem	nfnt 1702	If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
Theorem	nfnd 1703	Deduction associated with nfnt 1702. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
Theorem	nfn 1704	Inference associated with nfnt 1702. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 ¬ 𝜑
Theorem	nfdc 1705	If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥DECID 𝜑
Theorem	modal-5 1706	The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Theorem	19.9d 1707	A deduction version of one direction of 19.9 1690. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜓 → Ⅎ𝑥𝜑)    ⇒   ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))
Theorem	19.9hd 1708	A deduction version of one direction of 19.9 1690. This is an older variation of this theorem; new proofs should use 19.9d 1707. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜓 → (𝜑 → ∀𝑥𝜑))    ⇒   ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))
Theorem	excomim 1709	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
Theorem	excom 1710	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
Theorem	19.12 1711	Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	19.19 1712	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Theorem	19.21-2 1713	Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	nf2 1714	An alternate definition of df-nf 1507, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	nf3 1715	An alternate definition of df-nf 1507. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	nf4dc 1716	Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1717, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))
Theorem	nf4r 1717	If 𝜑 is always true or always false, then variable 𝑥 is effectively not free in 𝜑. The converse holds given a decidability condition, as seen at nf4dc 1716. (Contributed by Jim Kingdon, 21-Jul-2018.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)
Theorem	19.36i 1718	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ Ⅎ𝑥𝜓    &   ⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.36-1 1719	Closed form of 19.36i 1718. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.37-1 1720	One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37aiv 1721*	Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.38 1722	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	19.23t 1723	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23 1724	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.32dc 1725	Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))
Theorem	19.32r 1726	One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1725. (Contributed by Jim Kingdon, 28-Jul-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.31r 1727	One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.44 1728	Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
Theorem	19.45 1729	Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Theorem	19.34 1730	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
Theorem	19.41h 1731	Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1732 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41 1732	Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.42h 1733	Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1734 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	19.42 1734	Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	excom13 1735	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
Theorem	exrot3 1736	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
Theorem	exrot4 1737	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
Theorem	nexr 1738	Inference from 19.8a 1636. (Contributed by Jeff Hankins, 26-Jul-2009.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ ¬ 𝜑
Theorem	exan 1739	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥𝜑 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	hbexd 1740	Deduction form of bound-variable hypothesis builder hbex 1682. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))
Theorem	eeor 1741	Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
1.3.8  Equality theorems without distinct variables
Theorem	a9e 1742	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1493 through ax-14 2203 and ax-17 1572, all axioms other than ax-9 1577 are believed to be theorems of free logic, although the system without ax-9 1577 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	a9ev 1743*	At least one individual exists. Weaker version of a9e 1742. (Contributed by NM, 3-Aug-2017.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax9o 1744	An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	spimfv 1745*	Specialization, using implicit substitution. Version of spim 1784 with a disjoint variable condition. See spimv 1857 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	chvarfv 1746*	Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1803 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	equid 1747	Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms. This proof is similar to Tarski's and makes use of a dummy variable 𝑦. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)
⊢ 𝑥 = 𝑥
Theorem	nfequid 1748	Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
⊢ Ⅎ𝑦 𝑥 = 𝑥
Theorem	stdpc6 1749	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1816.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
⊢ ∀𝑥 𝑥 = 𝑥
Theorem	equcomi 1750	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	ax6evr 1751*	A commuted form of a9ev 1743. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)
⊢ ∃𝑥 𝑦 = 𝑥
Theorem	equcom 1752	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
Theorem	equcomd 1753	Deduction form of equcom 1752, symmetry of equality. For the versions for classes, see eqcom 2231 and eqcomd 2235. (Contributed by BJ, 6-Oct-2019.)
⊢ (𝜑 → 𝑥 = 𝑦)    ⇒   ⊢ (𝜑 → 𝑦 = 𝑥)
Theorem	equcoms 1754	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑦 = 𝑥 → 𝜑)
Theorem	equtr 1755	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
Theorem	equtrr 1756	A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
Theorem	equtr2 1757	A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
Theorem	equequ1 1758	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
Theorem	equequ2 1759	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))
Theorem	ax11i 1760	Inference that has ax-11 1552 (without ∀𝑦) as its conclusion and does not require ax-10 1551, ax-11 1552, or ax12 1558 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
1.3.9  Axioms ax-10 and ax-11
Theorem	ax10o 1761	Show that ax-10o 1762 can be derived from ax-10 1551. An open problem is whether this theorem can be derived from ax-10 1551 and the others when ax-11 1552 is replaced with ax-11o 1869. See Theorem ax10 1763 for the rederivation of ax-10 1551 from ax10o 1761. Normally, ax10o 1761 should be used rather than ax-10o 1762, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-10o 1762	Axiom ax-10o 1762 ("o" for "old") was the original version of ax-10 1551, before it was discovered (in May 2008) that the shorter ax-10 1551 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint). This axiom is redundant, as shown by Theorem ax10o 1761. Normally, ax10o 1761 should be used rather than ax-10o 1762, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	ax10 1763	Rederivation of ax-10 1551 from original version ax-10o 1762. See Theorem ax10o 1761 for the derivation of ax-10o 1762 from ax-10 1551. This theorem should not be referenced in any proof. Instead, use ax-10 1551 above so that uses of ax-10 1551 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	hbae 1764	All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	nfae 1765	All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
Theorem	hbaes 1766	Rule that applies hbae 1764 to antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑧∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
Theorem	hbnae 1767	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	nfnae 1768	All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
Theorem	hbnaes 1769	Rule that applies hbnae 1767 to antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
Theorem	naecoms 1770	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	equs4 1771	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	equsalh 1772	A useful equivalence related to substitution. New proofs should use equsal 1773 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsal 1773	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsex 1774	A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	equsexd 1775	Deduction form of equsex 1774. (Contributed by Jim Kingdon, 29-Dec-2017.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
Theorem	dral1 1776	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, 24-Nov-1994.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	dral2 1777	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	drex2 1778	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))
Theorem	drnf1 1779	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Theorem	drnf2 1780	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Theorem	spimth 1781	Closed theorem form of spim 1784. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	spimt 1782	Closed theorem form of spim 1784. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	spimh 1783	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1784 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spim 1784	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1784 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimeh 1785	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spimed 1786	Deduction version of spime 1787. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
⊢ (𝜒 → Ⅎ𝑥𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))
Theorem	spime 1787	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	cbv3 1788	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1790 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv3h 1789	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv3v 1790*	Rule used to change bound variables, using implicit substitution. Version of cbv3 1788 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv1 1791	Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv1h 1792	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv1v 1793*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv2h 1794	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbv2 1795	Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbv2w 1796*	Rule used to change bound variables, using implicit substitution. Version of cbv2 1795 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvalv1 1797*	Rule used to change bound variables, using implicit substitution. Version of cbval 1800 with a disjoint variable condition. See cbvalvw 1966 for a version with two disjoint variable conditions, and cbvalv 1964 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvexv1 1798*	Rule used to change bound variables, using implicit substitution. Version of cbvex 1802 with a disjoint variable condition. See cbvexvw 1967 for a version with two disjoint variable conditions, and cbvexv 1965 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbvalh 1799	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbval 1800	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)