P. 29 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 2801-2900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rmobidv 2801*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃*x ∈ A ψ ↔ ∃*x ∈ A χ))
Theorem	rmobiia 2802	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (x ∈ A → (φ ↔ ψ))    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ)
Theorem	rmobii 2803	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (φ ↔ ψ)    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ)
Theorem	raleqf 2804	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
Theorem	rexeqf 2805	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
Theorem	reueq1f 2806	Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))
Theorem	rmoeq1f 2807	Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B φ))
Theorem	raleq 2808*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
Theorem	rexeq 2809*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
Theorem	reueq1 2810*	Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))
Theorem	rmoeq1 2811*	Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B φ))
Theorem	raleqi 2812*	Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ A = B    ⇒   ⊢ (∀x ∈ A φ ↔ ∀x ∈ B φ)
Theorem	rexeqi 2813*	Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ A = B    ⇒   ⊢ (∃x ∈ A φ ↔ ∃x ∈ B φ)
Theorem	raleqdv 2814*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B ψ))
Theorem	rexeqdv 2815*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B ψ))
Theorem	raleqbi1dv 2816*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ))
Theorem	rexeqbi1dv 2817*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))
Theorem	reueqd 2818*	Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B ψ))
Theorem	rmoeqd 2819*	Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B ψ))
Theorem	raleqbidv 2820*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))
Theorem	rexeqbidv 2821*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))
Theorem	raleqbidva 2822*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (φ → A = B)    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))
Theorem	rexeqbidva 2823*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (φ → A = B)    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))
Theorem	mormo 2824	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*xφ → ∃*x ∈ A φ)
Theorem	reu5 2825	Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))
Theorem	reurex 2826	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
⊢ (∃!x ∈ A φ → ∃x ∈ A φ)
Theorem	reurmo 2827	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
⊢ (∃!x ∈ A φ → ∃*x ∈ A φ)
Theorem	rmo5 2828	Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
⊢ (∃*x ∈ A φ ↔ (∃x ∈ A φ → ∃!x ∈ A φ))
Theorem	nrexrmo 2829	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ (¬ ∃x ∈ A φ → ∃*x ∈ A φ)
Theorem	cbvralf 2830	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ ℲxA    &   ⊢ ℲyA    &   ⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)
Theorem	cbvrexf 2831	Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ ℲxA    &   ⊢ ℲyA    &   ⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)
Theorem	cbvral 2832*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)
Theorem	cbvrex 2833*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)
Theorem	cbvreu 2834*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)
Theorem	cbvrmo 2835*	Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*y ∈ A ψ)
Theorem	cbvralv 2836*	Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)
Theorem	cbvrexv 2837*	Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)
Theorem	cbvreuv 2838*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)
Theorem	cbvrmov 2839*	Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*y ∈ A ψ)
Theorem	cbvraldva2 2840*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((φ ∧ x = y) → (ψ ↔ χ))    &   ⊢ ((φ ∧ x = y) → A = B)    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀y ∈ B χ))
Theorem	cbvrexdva2 2841*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((φ ∧ x = y) → (ψ ↔ χ))    &   ⊢ ((φ ∧ x = y) → A = B)    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃y ∈ B χ))
Theorem	cbvraldva 2842*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((φ ∧ x = y) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀y ∈ A χ))
Theorem	cbvrexdva 2843*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((φ ∧ x = y) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃y ∈ A χ))
Theorem	cbvral2v 2844*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
⊢ (x = z → (φ ↔ χ))    &   ⊢ (y = w → (χ ↔ ψ))    ⇒   ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀z ∈ A ∀w ∈ B ψ)
Theorem	cbvrex2v 2845*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
⊢ (x = z → (φ ↔ χ))    &   ⊢ (y = w → (χ ↔ ψ))    ⇒   ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃z ∈ A ∃w ∈ B ψ)
Theorem	cbvral3v 2846*	Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
⊢ (x = w → (φ ↔ χ))    &   ⊢ (y = v → (χ ↔ θ))    &   ⊢ (z = u → (θ ↔ ψ))    ⇒   ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀w ∈ A ∀v ∈ B ∀u ∈ C ψ)
Theorem	cbvralsv 2847*	Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (∀x ∈ A φ ↔ ∀y ∈ A [y / x]φ)
Theorem	cbvrexsv 2848*	Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (∃x ∈ A φ ↔ ∃y ∈ A [y / x]φ)
Theorem	sbralie 2849*	Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀x ∈ y φ ↔ [y / x]∀y ∈ x ψ)
Theorem	rabbiia 2850	Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
⊢ (x ∈ A → (φ ↔ ψ))    ⇒   ⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ}
Theorem	rabbidva 2851*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
Theorem	rabbidv 2852*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
Theorem	rabeqf 2853	Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})
Theorem	rabeq 2854*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})
Theorem	rabeqbidv 2855*	Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ B ∣ χ})
Theorem	rabeqbidva 2856*	Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (φ → A = B)    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ B ∣ χ})
Theorem	reqabi 2857	Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
⊢ A = {x ∈ B ∣ φ}    ⇒   ⊢ (x ∈ A ↔ (x ∈ B ∧ φ))
Theorem	cbvrab 2858	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ ℲxA    &   ⊢ ℲyA    &   ⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ {x ∈ A ∣ φ} = {y ∈ A ∣ ψ}
Theorem	cbvrabv 2859*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ {x ∈ A ∣ φ} = {y ∈ A ∣ ψ}
2.1.6  The universal class
Syntax	cvv 2860	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 2861	Soundness justification theorem for df-v 2862. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {x ∣ x = x} = {y ∣ y = y}
Definition	df-v 2862	Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
⊢ V = {x ∣ x = x}
Theorem	vex 2863	All setvar variables are sets (see isset 2864). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
⊢ x ∈ V
Theorem	isset 2864*	Two ways to say "A is a set": A class A is a member of the universal class V (see df-v 2862) if and only if the class A exists (i.e. there exists some set x equal to class A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device "A ∈ V " to mean "A is a set" very frequently, for example in uniex 4318. Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg 4317, in order to shorten certain proofs we use the more general antecedent A ∈ V instead of A ∈ V to mean "A is a set." Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2349 requires that the expression substituted for B not contain x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 5-Aug-1993.)
⊢ (A ∈ V ↔ ∃x x = A)
Theorem	issetf 2865	A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ ℲxA    ⇒   ⊢ (A ∈ V ↔ ∃x x = A)
Theorem	isseti 2866*	A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
⊢ A ∈ V    ⇒   ⊢ ∃x x = A
Theorem	issetri 2867*	A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
⊢ ∃x x = A    ⇒   ⊢ A ∈ V
Theorem	elex 2868	If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (A ∈ B → A ∈ V)
Theorem	elexi 2869	If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
⊢ A ∈ B    ⇒   ⊢ A ∈ V
Theorem	elisset 2870*	An element of a class exists. (Contributed by NM, 1-May-1995.)
⊢ (A ∈ V → ∃x x = A)
Theorem	elex22 2871*	If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ ((A ∈ B ∧ A ∈ C) → ∃x(x ∈ B ∧ x ∈ C))
Theorem	elex2 2872*	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
⊢ (A ∈ B → ∃x x ∈ B)
Theorem	ralv 2873	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∀x ∈ V φ ↔ ∀xφ)
Theorem	rexv 2874	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∃x ∈ V φ ↔ ∃xφ)
Theorem	reuv 2875	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
⊢ (∃!x ∈ V φ ↔ ∃!xφ)
Theorem	rmov 2876	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*x ∈ V φ ↔ ∃*xφ)
Theorem	rabab 2877	A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ {x ∈ V ∣ φ} = {x ∣ φ}
Theorem	ralcom4 2878*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (∀x ∈ A ∀yφ ↔ ∀y∀x ∈ A φ)
Theorem	rexcom4 2879*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (∃x ∈ A ∃yφ ↔ ∃y∃x ∈ A φ)
Theorem	rexcom4a 2880*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ (∃x∃y ∈ A (φ ∧ ψ) ↔ ∃y ∈ A (φ ∧ ∃xψ))
Theorem	rexcom4b 2881*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ B ∈ V    ⇒   ⊢ (∃x∃y ∈ A (φ ∧ x = B) ↔ ∃y ∈ A φ)
Theorem	ceqsalt 2882*	Closed theorem version of ceqsalg 2884. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) ↔ ψ))
Theorem	ceqsralt 2883*	Restricted quantifier version of ceqsalt 2882. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ ψ))
Theorem	ceqsalg 2884*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎxψ    &   ⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (A ∈ V → (∀x(x = A → φ) ↔ ψ))
Theorem	ceqsal 2885*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎxψ    &   ⊢ A ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (∀x(x = A → φ) ↔ ψ)
Theorem	ceqsalv 2886*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
⊢ A ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (∀x(x = A → φ) ↔ ψ)
Theorem	ceqsralv 2887*	Restricted quantifier version of ceqsalv 2886. (Contributed by NM, 21-Jun-2013.)
⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (A ∈ B → (∀x ∈ B (x = A → φ) ↔ ψ))
Theorem	gencl 2888*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (θ ↔ ∃x(χ ∧ A = B))    &   ⊢ (A = B → (φ ↔ ψ))    &   ⊢ (χ → φ)    ⇒   ⊢ (θ → ψ)
Theorem	2gencl 2889*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (C ∈ S ↔ ∃x ∈ R A = C)    &   ⊢ (D ∈ S ↔ ∃y ∈ R B = D)    &   ⊢ (A = C → (φ ↔ ψ))    &   ⊢ (B = D → (ψ ↔ χ))    &   ⊢ ((x ∈ R ∧ y ∈ R) → φ)    ⇒   ⊢ ((C ∈ S ∧ D ∈ S) → χ)
Theorem	3gencl 2890*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (D ∈ S ↔ ∃x ∈ R A = D)    &   ⊢ (F ∈ S ↔ ∃y ∈ R B = F)    &   ⊢ (G ∈ S ↔ ∃z ∈ R C = G)    &   ⊢ (A = D → (φ ↔ ψ))    &   ⊢ (B = F → (ψ ↔ χ))    &   ⊢ (C = G → (χ ↔ θ))    &   ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → φ)    ⇒   ⊢ ((D ∈ S ∧ F ∈ S ∧ G ∈ S) → θ)
Theorem	cgsexg 2891*	Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
⊢ (x = A → χ)    &   ⊢ (χ → (φ ↔ ψ))    ⇒   ⊢ (A ∈ V → (∃x(χ ∧ φ) ↔ ψ))
Theorem	cgsex2g 2892*	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
⊢ ((x = A ∧ y = B) → χ)    &   ⊢ (χ → (φ ↔ ψ))    ⇒   ⊢ ((A ∈ V ∧ B ∈ W) → (∃x∃y(χ ∧ φ) ↔ ψ))
Theorem	cgsex4g 2893*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → χ)    &   ⊢ (χ → (φ ↔ ψ))    ⇒   ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(χ ∧ φ) ↔ ψ))
Theorem	ceqsex 2894*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎxψ    &   ⊢ A ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (∃x(x = A ∧ φ) ↔ ψ)
Theorem	ceqsexv 2895*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
⊢ A ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (∃x(x = A ∧ φ) ↔ ψ)
Theorem	ceqsex2 2896*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
⊢ Ⅎxψ    &   ⊢ Ⅎyχ    &   ⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    &   ⊢ (y = B → (ψ ↔ χ))    ⇒   ⊢ (∃x∃y(x = A ∧ y = B ∧ φ) ↔ χ)
Theorem	ceqsex2v 2897*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    &   ⊢ (y = B → (ψ ↔ χ))    ⇒   ⊢ (∃x∃y(x = A ∧ y = B ∧ φ) ↔ χ)
Theorem	ceqsex3v 2898*	Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    &   ⊢ (y = B → (ψ ↔ χ))    &   ⊢ (z = C → (χ ↔ θ))    ⇒   ⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ θ)
Theorem	ceqsex4v 2899*	Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    &   ⊢ (y = B → (ψ ↔ χ))    &   ⊢ (z = C → (χ ↔ θ))    &   ⊢ (w = D → (θ ↔ τ))    ⇒   ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ τ)
Theorem	ceqsex6v 2900*	Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    &   ⊢ E ∈ V    &   ⊢ F ∈ V    &   ⊢ (x = A → (φ ↔ ψ))    &   ⊢ (y = B → (ψ ↔ χ))    &   ⊢ (z = C → (χ ↔ θ))    &   ⊢ (w = D → (θ ↔ τ))    &   ⊢ (v = E → (τ ↔ η))    &   ⊢ (u = F → (η ↔ ζ))    ⇒   ⊢ (∃x∃y∃z∃w∃v∃u((x = A ∧ y = B ∧ z = C) ∧ (w = D ∧ v = E ∧ u = F) ∧ φ) ↔ ζ)