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Theorem List for New Foundations Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcbii 3101 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(φψ)       ([.A / x].φ ↔ [.A / x].ψ)
 
TheoremsbcbiiOLD 3102 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)
(φψ)       (A V → ([.A / x].φ ↔ [.A / x].ψ))
 
Theoremeqsbc3r 3103* eqsbc3 3085 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD (future) using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
(A B → ([.A / x].C = xC = A))
 
Theoremsbc3ang 3104 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].(φ ψ χ) ↔ ([.A / x].φ [.A / x].ψ [.A / x].χ)))
 
Theoremsbcel1gv 3105* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].x BA B))
 
Theoremsbcel2gv 3106* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(B V → ([.B / x].A xA B))
 
Theoremsbcimdv 3107* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
(φ → (ψχ))       ((φ A V) → ([.A / x].ψ → [.A / x].χ))
 
Theoremsbctt 3108 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
((A V xφ) → ([.A / x].φφ))
 
Theoremsbcgf 3109 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
xφ       (A V → ([.A / x].φφ))
 
Theoremsbc19.21g 3110 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
xφ       (A V → ([.A / x].(φψ) ↔ (φ → [.A / x].ψ)))
 
Theoremsbcg 3111* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3109. (Contributed by Alan Sare, 10-Nov-2012.)
(A V → ([.A / x].φφ))
 
Theoremsbc2iegf 3112* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
xψ    &   yψ    &   x B W    &   ((x = A y = B) → (φψ))       ((A V B W) → ([.A / x].[.B / y].φψ))
 
Theoremsbc2ie 3113* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
A V    &   B V    &   ((x = A y = B) → (φψ))       ([.A / x].[.B / y].φψ)
 
Theoremsbc2iedv 3114* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
A V    &   B V    &   (φ → ((x = A y = B) → (ψχ)))       (φ → ([.A / x].[.B / y].ψχ))
 
Theoremsbc3ie 3115* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
A V    &   B V    &   C V    &   ((x = A y = B z = C) → (φψ))       ([.A / x].[.B / y].[.C / z].φψ)
 
Theoremsbccomlem 3116* Lemma for sbccom 3117. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([.A / x].[.B / y].φ ↔ [.B / y].[.A / x].φ)
 
Theoremsbccom 3117* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
([.A / x].[.B / y].φ ↔ [.B / y].[.A / x].φ)
 
Theoremsbcralt 3118* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
((A V F/_yA) → ([.A / x].y B φy B [.A / x].φ))
 
Theoremsbcrext 3119* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((A V F/_yA) → ([.A / x].y B φy B [.A / x].φ))
 
Theoremsbcralg 3120* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].y B φy B [.A / x].φ))
 
Theoremsbcrexg 3121* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].y B φy B [.A / x].φ))
 
Theoremsbcreug 3122* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(A V → ([.A / x].∃!y B φ∃!y B [.A / x].φ))
 
Theoremsbcabel 3123* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
F/_xB       (A V → ([.A / x].{y φ} B ↔ {y [.A / x].φ} B))
 
Theoremrspsbc 3124* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2024 and spsbc 3058. See also rspsbca 3125 and rspcsbela 3195. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(A B → (x B φ → [.A / x].φ))
 
Theoremrspsbca 3125* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((A B x B φ) → [.A / x].φ)
 
Theoremrspesbca 3126* Existence form of rspsbca 3125. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((A B [.A / x].φ) → x B φ)
 
Theoremspesbc 3127 Existence form of spsbc 3058. (Contributed by Mario Carneiro, 18-Nov-2016.)
([.A / x].φxφ)
 
Theoremspesbcd 3128 form of spsbc 3058. (Contributed by Mario Carneiro, 9-Feb-2017.)
(φ → [.A / x].ψ)       (φxψ)
 
Theoremsbcth2 3129* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(x Bφ)       (A B → [.A / x].φ)
 
Theoremra5 3130 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1798. (Contributed by NM, 16-Jan-2004.)
xφ       (x A (φψ) → (φx A ψ))
 
Theoremrmo2 3131* Alternate definition of restricted "at most one." Note that ∃*x Aφ is not equivalent to y Ax A(φx = y) (in analogy to reu6 3025); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3132. (Contributed by NM, 17-Jun-2017.)
yφ       (∃*x Aφyx A (φx = y))
 
Theoremrmo2i 3132* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
yφ       (y A x A (φx = y) → ∃*x Aφ)
 
Theoremrmo3 3133* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
yφ       (∃*x Aφx A y A ((φ [y / x]φ) → x = y))
 
Theoremrmob 3134* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
(x = B → (φψ))    &   (x = C → (φχ))       ((∃*x Aφ (B A ψ)) → (B = C ↔ (C A χ)))
 
Theoremrmoi 3135* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
(x = B → (φψ))    &   (x = C → (φχ))       ((∃*x Aφ (B A ψ) (C A χ)) → B = C)
 
2.1.9  Proper substitution of classes for sets into classes
 
Syntaxcsb 3136 Extend class notation to include the proper substitution of a class for a set into another class.
class [A / x]B
 
Definitiondf-csb 3137* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3046, to prevent ambiguity. Theorem sbcel1g 3155 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3164 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
[A / x]B = {y [.A / x].y B}
 
Theoremcsb2 3138* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)
[A / x]B = {y x(x = A y B)}
 
Theoremcsbeq1 3139 Analog of dfsbcq 3048 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(A = B[A / x]C = [B / x]C)
 
Theoremcbvcsb 3140 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
F/_yC    &   F/_xD    &   (x = yC = D)       [A / x]C = [A / y]D
 
Theoremcbvcsbv 3141* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
(x = yB = C)       [A / x]B = [A / y]C
 
Theoremcsbeq1d 3142 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(φA = B)       (φ[A / x]C = [B / x]C)
 
Theoremcsbid 3143 Analog of sbid 1922 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
[x / x]A = A
 
Theoremcsbeq1a 3144 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(x = AB = [A / x]B)
 
Theoremcsbco 3145* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
[A / y][y / x]B = [A / x]B
 
Theoremcsbexg 3146 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
((A V x B W) → [A / x]B V)
 
Theoremcsbex 3147 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
A V    &   B V       [A / x]B V
 
Theoremcsbtt 3148 Substitution doesn't affect a constant B (in which x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
((A V F/_xB) → [A / x]B = B)
 
Theoremcsbconstgf 3149 Substitution doesn't affect a constant B (in which x is not free). (Contributed by NM, 10-Nov-2005.)
F/_xB       (A V[A / x]B = B)
 
Theoremcsbconstg 3150* Substitution doesn't affect a constant B (in which x is not free). csbconstgf 3149 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(A V[A / x]B = B)
 
Theoremsbcel12g 3151 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].B C[A / x]B [A / x]C))
 
Theoremsbceqg 3152 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A V → ([.A / x].B = C[A / x]B = [A / x]C))
 
Theoremsbcnel12g 3153 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(A V → ([.A / x].B C[A / x]B [A / x]C))
 
Theoremsbcne12g 3154 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
(A V → ([.A / x].BC[A / x]B[A / x]C))
 
Theoremsbcel1g 3155* Move proper substitution in and out of a membership relation. Note that the scope of [.A / x]. is the wff B C, whereas the scope of [A / x] is the class B. (Contributed by NM, 10-Nov-2005.)
(A V → ([.A / x].B C[A / x]B C))
 
Theoremsbceq1g 3156* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(A V → ([.A / x].B = C[A / x]B = C))
 
Theoremsbcel2g 3157* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
(A V → ([.A / x].B CB [A / x]C))
 
Theoremsbceq2g 3158* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(A V → ([.A / x].B = CB = [A / x]C))
 
Theoremcsbcomg 3159* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
((A V B W) → [A / x][B / y]C = [B / y][A / x]C)
 
Theoremcsbeq2d 3160 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
xφ    &   (φB = C)       (φ[A / x]B = [A / x]C)
 
Theoremcsbeq2dv 3161* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
(φB = C)       (φ[A / x]B = [A / x]C)
 
Theoremcsbeq2i 3162 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
B = C       [A / x]B = [A / x]C
 
Theoremcsbvarg 3163 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
(A V[A / x]x = A)
 
Theoremsbccsbg 3164* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
(A V → ([.A / x].φy [A / x]{y φ}))
 
Theoremsbccsb2g 3165 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
(A V → ([.A / x].φA [A / x]{x φ}))
 
Theoremnfcsb1d 3166 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(φ → F/_xA)       (φ → F/_x[A / x]B)
 
Theoremnfcsb1 3167 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
F/_xA       F/_x[A / x]B
 
Theoremnfcsb1v 3168* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
F/_x[A / x]B
 
Theoremnfcsbd 3169 Deduction version of nfcsb 3170. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
yφ    &   (φ → F/_xA)    &   (φ → F/_xB)       (φ → F/_x[A / y]B)
 
Theoremnfcsb 3170 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
F/_xA    &   F/_xB       F/_x[A / y]B
 
Theoremcsbhypf 3171* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2904 for class substitution version. (Contributed by NM, 19-Dec-2008.)
F/_xA    &   F/_xC    &   (x = AB = C)       (y = A[y / x]B = C)
 
Theoremcsbiebt 3172* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3176.) (Contributed by NM, 11-Nov-2005.)
((A V F/_xC) → (x(x = AB = C) ↔ [A / x]B = C))
 
Theoremcsbiedf 3173* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
xφ    &   (φ → F/_xC)    &   (φA V)    &   ((φ x = A) → B = C)       (φ[A / x]B = C)
 
Theoremcsbieb 3174* Bidirectional conversion between an implicit class substitution hypothesis x = AB = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
A V    &   F/_xC       (x(x = AB = C) ↔ [A / x]B = C)
 
Theoremcsbiebg 3175* Bidirectional conversion between an implicit class substitution hypothesis x = AB = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
F/_xC       (A V → (x(x = AB = C) ↔ [A / x]B = C))
 
Theoremcsbiegf 3176* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
(A V → F/_xC)    &   (x = AB = C)       (A V[A / x]B = C)
 
Theoremcsbief 3177* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
A V    &   F/_xC    &   (x = AB = C)       [A / x]B = C
 
Theoremcsbied 3178* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
(φA V)    &   ((φ x = A) → B = C)       (φ[A / x]B = C)
 
Theoremcsbied2 3179* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φA V)    &   (φA = B)    &   ((φ x = B) → C = D)       (φ[A / x]C = D)
 
Theoremcsbie2t 3180* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3181). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
A V    &   B V       (xy((x = A y = B) → C = D) → [A / x][B / y]C = D)
 
Theoremcsbie2 3181* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
A V    &   B V    &   ((x = A y = B) → C = D)       [A / x][B / y]C = D
 
Theoremcsbie2g 3182* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
(x = yB = C)    &   (y = AC = D)       (A V[A / x]B = D)
 
Theoremsbcnestgf 3183 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
((A V yxφ) → ([.A / x].[.B / y].φ ↔ [.[A / x]B / y].φ))
 
Theoremcsbnestgf 3184 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
((A V yF/_xC) → [A / x][B / y]C = [[A / x]B / y]C)
 
Theoremsbcnestg 3185* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(A V → ([.A / x].[.B / y].φ ↔ [.[A / x]B / y].φ))
 
Theoremcsbnestg 3186* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
(A V[A / x][B / y]C = [[A / x]B / y]C)
 
TheoremcsbnestgOLD 3187* Nest the composition of two substitutions. (New usage is discouraged.) (Contributed by NM, 23-Nov-2005.)
((A V x B W) → [A / x][B / y]C = [[A / x]B / y]C)
 
Theoremcsbnest1g 3188 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(A V[A / x][B / x]C = [[A / x]B / x]C)
 
Theoremcsbnest1gOLD 3189* Nest the composition of two substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((A V x B W) → [A / x][B / x]C = [[A / x]B / x]C)
 
Theoremcsbidmg 3190* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
(A V[A / x][A / x]B = [A / x]B)
 
Theoremsbcco3g 3191* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(x = AB = C)       (A V → ([.A / x].[.B / y].φ ↔ [.C / y].φ))
 
Theoremsbcco3gOLD 3192* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
(x = AB = C)       ((A V x B W) → ([.A / x].[.B / y].φ ↔ [.C / y].φ))
 
Theoremcsbco3g 3193* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(x = AB = C)       (A V[A / x][B / y]D = [C / y]D)
 
Theoremcsbco3gOLD 3194* Composition of two class substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
(x = AB = D)       ((A V x B W) → [A / x][B / y]C = [D / y]C)
 
Theoremrspcsbela 3195* Special case related to rspsbc 3124. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((A B x B C D) → [A / x]C D)
 
Theoremsbnfc2 3196* Two ways of expressing "x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
(F/_xAyz[y / x]A = [z / x]A)
 
Theoremcsbabg 3197* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A V[A / x]{y φ} = {y [.A / x].φ})
 
Theoremcbvralcsf 3198 A more general version of cbvralf 2829 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
F/_yA    &   F/_xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (x A φy B ψ)
 
Theoremcbvrexcsf 3199 A more general version of cbvrexf 2830 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
F/_yA    &   F/_xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (x A φy B ψ)
 
Theoremcbvreucsf 3200 A more general version of cbvreuv 2837 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
F/_yA    &   F/_xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (∃!x A φ∃!y B ψ)
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