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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 / xφ ↔ [̣A / xψ)

TheoremsbcbiiOLD 3102 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)
(φψ)       (A V → ([̣A / xφ ↔ [̣A / xψ))

Theoremeqsbc3r 3103* eqsbc3 3085 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD in set.mm using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
(A B → ([̣A / xC = 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 / xφ A / xψ A / xχ)))

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 / xx 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 / xA 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 / xψ → [̣A / xχ))

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 / xφφ))

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 / xφφ))

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 / xψ)))

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 / xφφ))

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 / yφψ))

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 / yφψ)

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 / yψχ))

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 / zφψ)

Theoremsbccomlem 3116* Lemma for sbccom 3117. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([̣A / x]̣[̣B / yφ ↔ [̣B / y]̣[̣A / xφ)

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 / yφ ↔ [̣B / y]̣[̣A / xφ)

Theoremsbcralt 3118* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
((A V yA) → ([̣A / xy B φy BA / xφ))

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 yA) → ([̣A / xy B φy BA / xφ))

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 / xy B φy BA / xφ))

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 / xy B φy BA / xφ))

Theoremsbcreug 3122* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(A V → ([̣A / x∃!y B φ∃!y BA / xφ))

Theoremsbcabel 3123* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
xB       (A V → ([̣A / x]̣{y φ} B ↔ {y A / xφ} 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 / xφ))

Theoremrspsbca 3125* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((A B x B φ) → [̣A / xφ)

Theoremrspesbca 3126* Existence form of rspsbca 3125. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((A B A / xφ) → x B φ)

Theoremspesbc 3127 Existence form of spsbc 3058. (Contributed by Mario Carneiro, 18-Nov-2016.)
([̣A / xφxφ)

Theoremspesbcd 3128 form of spsbc 3058. (Contributed by Mario Carneiro, 9-Feb-2017.)
(φ → [̣A / xψ)       (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 / xφ)

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 A φyx 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 A φ)

Theoremrmo3 3133* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
yφ       (∃*x A φx 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 A φ (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 A φ (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 / xy 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.)
yC    &   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 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.)
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 / xB 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 / xB = 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 / xB 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 / xBC[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 / xB 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 / xB = 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 / xB CB [A / x]C))

Theoremsbceq2g 3158* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(A V → ([̣A / xB = 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 setvar 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 / xφy [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 / xφA [A / x]{x φ}))

Theoremnfcsb1d 3166 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(φxA)       (φx[A / x]B)

Theoremnfcsb1 3167 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
xA       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.)
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    &   (φxA)    &   (φxB)       (φx[A / y]B)

Theoremnfcsb 3170 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
xA    &   xB       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.)
xA    &   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 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    &   (φxC)    &   (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    &   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.)
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 VxC)    &   (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    &   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 / yφ ↔ [̣[A / x]B / yφ))

Theoremcsbnestgf 3184 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
((A V yxC) → [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 / yφ ↔ [̣[A / x]B / yφ))

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 / yφ ↔ [̣C / yφ))

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 / yφ ↔ [̣C / yφ))

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.)
(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 / xφ})

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.)
yA    &   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.)
yA    &   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.)
yA    &   xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (∃!x A φ∃!y B ψ)

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