P. 34 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	spc2ev 3301*	Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ps -> E. x E. y ph )
Theorem	rspct 3302*	A closed version of rspc 3303. (Contributed by Andrew Salmon, 6-Jun-2011.)
|- F/ x ps   =>    |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )
Theorem	rspc 3303*	Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ph -> ps ) )
Theorem	rspce 3304*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps ) -> E. x e. B ph )
Theorem	rspcv 3305*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ph -> ps ) )
Theorem	rspccv 3306*	Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )
Theorem	rspcva 3307*	Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ A. x e. B ph ) -> ps )
Theorem	rspccva 3308*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A. x e. B ph /\ A e. B ) -> ps )
Theorem	rspcev 3309*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps ) -> E. x e. B ph )
Theorem	rspcimdv 3310*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x e. B ps -> ch ) )
Theorem	rspcimedv 3311*	Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ch -> ps ) )   =>    |- ( ph -> ( ch -> E. x e. B ps ) )
Theorem	rspcdv 3312*	Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps -> ch ) )
Theorem	rspcedv 3313*	Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch -> E. x e. B ps ) )
Theorem	rspcebdv 3314*	Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ( ph /\ ps ) -> x = A )   =>    |- ( ph -> ( E. x e. B ps <-> ch ) )
Theorem	rspcda 3315*	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)
|- ( x = C -> ( ps <-> ch ) )   &    |- ( ph -> A. x e. A ps )   &    |- ( ph -> C e. A )   &    |- F/ x ph   =>    |- ( ph -> ch )
Theorem	rspcdva 3316*	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
|- ( x = C -> ( ps <-> ch ) )   &    |- ( ph -> A. x e. A ps )   &    |- ( ph -> C e. A )   =>    |- ( ph -> ch )
Theorem	rspcedvd 3317*	Restricted existential specialization, using implicit substitution. Variant of rspcedv 3313. (Contributed by AV, 27-Nov-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ch )   =>    |- ( ph -> E. x e. B ps )
Theorem	rspcedeq1vd 3318*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3317 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> C = D )   =>    |- ( ph -> E. x e. B C = D )
Theorem	rspcedeq2vd 3319*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3317 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> C = D )   =>    |- ( ph -> E. x e. B C = D )
Theorem	rspc2 3320*	Restricted specialization with two quantifiers, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
|- F/ x ch   &    |- F/ y ps   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
Theorem	rspc2gv 3321*	Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x e. V A. y e. W ph -> ps ) )
Theorem	rspc2v 3322*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
Theorem	rspc2va 3323*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) -> ps )
Theorem	rspc2ev 3324*	2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   =>    |- ( ( A e. C /\ B e. D /\ ps ) -> E. x e. C E. y e. D ph )
Theorem	rspc3v 3325*	3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> th ) )   &    |- ( z = C -> ( th <-> ps ) )   =>    |- ( ( A e. R /\ B e. S /\ C e. T ) -> ( A. x e. R A. y e. S A. z e. T ph -> ps ) )
Theorem	rspc3ev 3326*	3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( y = B -> ( ch <-> th ) )   &    |- ( z = C -> ( th <-> ps ) )   =>    |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) -> E. x e. R E. y e. S E. z e. T ph )
Theorem	ralxpxfr2d 3327*	Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- A e. _V   &    |- ( ph -> ( x e. B <-> E. y e. C E. z e. D x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C A. z e. D ch ) )
Theorem	rexraleqim 3328*	Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
|- ( x = z -> ( ps <-> ph ) )   &    |- ( z = Y -> ( ph <-> th ) )   =>    |- ( ( E. z e. A ph /\ A. x e. A ( ps -> x = Y ) ) -> th )
Theorem	eqvincg 3329*	A variable introduction law for class equality, closed form. (Contributed by Thierry Arnoux, 2-Mar-2017.)
|- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )
Theorem	eqvinc 3330*	A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.)
|- A e. _V   =>    |- ( A = B <-> E. x ( x = A /\ x = B ) )
Theorem	eqvincf 3331	A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
|- F/_ x A   &    |- F/_ x B   &    |- A e. _V   =>    |- ( A = B <-> E. x ( x = A /\ x = B ) )
Theorem	alexeqg 3332*	Two ways to express substitution of A for x in ph. This is the analogue for classes of sb56 2150. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
|- ( A e. V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
Theorem	ceqex 3333*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)
|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
Theorem	ceqsexg 3334*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsexgv 3335*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsrexv 3336*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
Theorem	ceqsrexbv 3337*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )
Theorem	ceqsrex2v 3338*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )
Theorem	clel2 3339*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- A e. _V   =>    |- ( A e. B <-> A. x ( x = A -> x e. B ) )
Theorem	clel3g 3340*	An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )
Theorem	clel3 3341*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- B e. _V   =>    |- ( A e. B <-> E. x ( x = B /\ A e. x ) )
Theorem	clel4 3342*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
|- B e. _V   =>    |- ( A e. B <-> A. x ( x = B -> A e. x ) )
Theorem	clel5 3343*	Alternate definition of class membership: a class X is an element of another class A iff there is an element of A equal to X. (Contributed by AV, 13-Nov-2020.)
|- ( X e. A <-> E. x e. A X = x )
Theorem	pm13.183 3344*	Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) )
Theorem	rr19.3v 3345*	Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4064 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
|- ( A. x e. A A. y e. A ph <-> A. x e. A ph )
Theorem	rr19.28v 3346*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4066 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> A. x e. A ( ph /\ A. y e. A ps ) )
Theorem	elabgt 3347*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3351.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elabgf 3348	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A e. { x | ph } <-> ps ) )
Theorem	elabf 3349*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elab 3350*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elabg 3351*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
Theorem	elabd 3352*	Explicit demonstration the class { x | ps } is not empty by the example X. (Contributed by RP, 12-Aug-2020.)
|- ( ph -> X e. _V )   &    |- ( ph -> ch )   &    |- ( x = X -> ( ps <-> ch ) )   =>    |- ( ph -> E. x ps )
Theorem	elab2g 3353*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. V -> ( A e. B <-> ps ) )
Theorem	elab2 3354*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. B <-> ps )
Theorem	elab4g 3355*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
|- ( x = A -> ( ph <-> ps ) )   &    |- B = { x | ph }   =>    |- ( A e. B <-> ( A e. _V /\ ps ) )
Theorem	elab3gf 3356	Membership in a class abstraction, with a weaker antecedent than elabgf 3348. (Contributed by NM, 6-Sep-2011.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3g 3357*	Membership in a class abstraction, with a weaker antecedent than elabg 3351. (Contributed by NM, 29-Aug-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )
Theorem	elab3 3358*	Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
|- ( ps -> A e. _V )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x | ph } <-> ps )
Theorem	elrabi 3359*	Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
|- ( A e. { x e. V | ph } -> A e. V )
Theorem	elrabf 3360	Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
Theorem	rabtru 3361	Abstract builder using the constant wff T. (Contributed by Thierry Arnoux, 4-May-2020.)
|- F/_ x A   =>    |- { x e. A | T. } = A
Theorem	elrab3t 3362*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3364.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
|- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )
Theorem	elrab 3363*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
Theorem	elrab3 3364*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )
Theorem	elrabd 3365*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3363. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( x = A -> ( ps <-> ch ) )   &    |- ( ph -> A e. B )   &    |- ( ph -> ch )   =>    |- ( ph -> A e. { x e. B | ps } )
Theorem	elrab2 3366*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- C = { x e. B | ph }   =>    |- ( A e. C <-> ( A e. B /\ ps ) )
Theorem	ralab 3367*	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )
Theorem	ralrab 3368*	Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )
Theorem	rexab 3369*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )
Theorem	rexrab 3370*	Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
|- ( y = x -> ( ph <-> ps ) )   =>    |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )
Theorem	ralab2 3371*	Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )
Theorem	ralrab2 3372*	Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )
Theorem	rexab2 3373*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) )
Theorem	rexrab2 3374*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( x = y -> ( ps <-> ch ) )   =>    |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )
Theorem	abidnf 3375*	Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
|- ( F/_ x A -> { z | A. x z e. A } = A )
Theorem	dedhb 3376*	A deduction theorem for converting the inference |- F/_ x A => |- ph into a closed theorem. Use nfa1 2028 and nfab 2769 to eliminate the hypothesis of the substitution instance ps of the inference. For converting the inference form into a deduction form, abidnf 3375 is useful. (Contributed by NM, 8-Dec-2006.)
|- ( A = { z | A. x z e. A } -> ( ph <-> ps ) )   &    |- ps   =>    |- ( F/_ x A -> ph )
Theorem	eqeu 3377*	A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph )
Theorem	eueq 3378*	Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
|- ( A e. _V <-> E! x x = A )
Theorem	eueq1 3379*	Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- A e. _V   =>    |- E! x x = A
Theorem	eueq2 3380*	Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
|- A e. _V   &    |- B e. _V   =>    |- E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) )
Theorem	eueq3 3381*	Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- -. ( ph /\ ps )   =>    |- E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )
Theorem	moeq 3382*	There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
|- E* x x = A
Theorem	moeq3 3383*	"At most one" property of equality (split into 3 cases). (The first two hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)
|- B e. _V   &    |- C e. _V   &    |- -. ( ph /\ ps )   =>    |- E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )
Theorem	mosub 3384*	"At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
|- E* x ph   =>    |- E* x E. y ( y = A /\ ph )
Theorem	mo2icl 3385*	Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
|- ( A. x ( ph -> x = A ) -> E* x ph )
Theorem	mob2 3386*	Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
Theorem	moi2 3387*	Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A )
Theorem	mob 3388*	Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
Theorem	moi 3389*	Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )
Theorem	morex 3390*	Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- B e. _V   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) )
Theorem	euxfr2 3391*	Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
|- A e. _V   &    |- E* y x = A   =>    |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph )
Theorem	euxfr 3392*	Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
|- A e. _V   &    |- E! y x = A   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E! x ph <-> E! y ps )
Theorem	euind 3393*	Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
|- B e. _V   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x = y -> A = B )   =>    |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph ) -> E! z A. x ( ph -> z = A ) )
Theorem	reu2 3394*	A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
|- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
Theorem	reu6 3395*	A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
|- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
Theorem	reu3 3396*	A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
|- ( E! x e. A ph <-> ( E. x e. A ph /\ E. y e. A A. x e. A ( ph -> x = y ) ) )
Theorem	reu6i 3397*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
Theorem	eqreu 3398*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )
Theorem	rmo4 3399*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
Theorem	reu4 3400*	Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) )