P. 33 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vjust 3201	Soundness justification theorem for df-v 3202. (Contributed by Rodolfo Medina, 27-Apr-2010.)
|- { x | x = x } = { y | y = y }
Definition	df-v 3202	Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. The class _V can be described as the "class of all sets"; vprc 4796 proves that _V is not itself a set in ZFC. We will frequently use the expression A e. _V as a short way to say " A is a set", and isset 3207 proves that this expression has the same meaning as E. x x = A. The class _V is called the "von Neumann universe", however, the letter "V" is not a tribute to the name of von Neumann. The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the word "Verum". Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)
|- _V = { x | x = x }
Theorem	vex 3203	All setvar variables are sets (see isset 3207). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.)
|- x e. _V
Theorem	eqvf 3204	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
|- F/_ x A   =>    |- ( A = _V <-> A. x x e. A )
Theorem	eqv 3205*	The universe contains every set. (Contributed by NM, 11-Sep-2006.)
|- ( A = _V <-> A. x x e. A )
Theorem	abv 3206	The class of sets verifying a property is the universal class if and only if that property is a tautology. (Contributed by BJ, 19-Mar-2021.)
|- ( { x | ph } = _V <-> A. x ph )
Theorem	isset 3207*	Two ways to say " A is a set": A class A is a member of the universal class _V (see df-v 3202) 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 e. _V " to mean " A is a set" very frequently, for example in uniex 6953. Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg 6955, in order to shorten certain proofs we use the more general antecedent A e. V instead of A e. _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 2618 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, 26-May-1993.)
|- ( A e. _V <-> E. x x = A )
Theorem	issetf 3208	A version of isset 3207 that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- ( A e. _V <-> E. x x = A )
Theorem	isseti 3209*	A way to say " A is a set" (inference rule). (Contributed by NM, 24-Jun-1993.)
|- A e. _V   =>    |- E. x x = A
Theorem	issetri 3210*	A way to say " A is a set" (inference rule). (Contributed by NM, 21-Jun-1993.)
|- E. x x = A   =>    |- A e. _V
Theorem	eqvisset 3211	A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 3207 and issetri 3210. (Contributed by BJ, 27-Apr-2019.)
|- ( x = A -> A e. _V )
Theorem	elex 3212	If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A e. B -> A e. _V )
Theorem	elexi 3213	If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
|- A e. B   =>    |- A e. _V
Theorem	elexd 3214	If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   =>    |- ( ph -> A e. _V )
Theorem	elisset 3215*	An element of a class exists. (Contributed by NM, 1-May-1995.)
|- ( A e. V -> E. x x = A )
Theorem	elex2 3216*	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
|- ( A e. B -> E. x x e. B )
Theorem	elex22 3217*	If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )
Theorem	prcnel 3218	A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.)
|- ( -. A e. _V -> -. A e. V )
Theorem	ralv 3219	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- ( A. x e. _V ph <-> A. x ph )
Theorem	rexv 3220	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
|- ( E. x e. _V ph <-> E. x ph )
Theorem	reuv 3221	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
|- ( E! x e. _V ph <-> E! x ph )
Theorem	rmov 3222	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- ( E* x e. _V ph <-> E* x ph )
Theorem	rabab 3223	A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- { x e. _V | ph } = { x | ph }
Theorem	ralcom4 3224*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )
Theorem	rexcom4 3225*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )
Theorem	rexcom4a 3226*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
Theorem	rexcom4b 3227*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
|- B e. _V   =>    |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )
Theorem	ceqsalt 3228*	Closed theorem version of ceqsalg 3230. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )
Theorem	ceqsralt 3229*	Restricted quantifier version of ceqsalt 3228. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
Theorem	ceqsalg 3230*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3231. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )
Theorem	ceqsalgALT 3231*	Alternate proof of ceqsalg 3230, not using ceqsalt 3228. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )
Theorem	ceqsal 3232*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ( x = A -> ph ) <-> ps )
Theorem	ceqsalv 3233*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ( x = A -> ph ) <-> ps )
Theorem	ceqsralv 3234*	Restricted quantifier version of ceqsalv 3233. (Contributed by NM, 21-Jun-2013.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )
Theorem	gencl 3235*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( th <-> E. x ( ch /\ A = B ) )   &    |- ( A = B -> ( ph <-> ps ) )   &    |- ( ch -> ph )   =>    |- ( th -> ps )
Theorem	2gencl 3236*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( C e. S <-> E. x e. R A = C )   &    |- ( D e. S <-> E. y e. R B = D )   &    |- ( A = C -> ( ph <-> ps ) )   &    |- ( B = D -> ( ps <-> ch ) )   &    |- ( ( x e. R /\ y e. R ) -> ph )   =>    |- ( ( C e. S /\ D e. S ) -> ch )
Theorem	3gencl 3237*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
|- ( D e. S <-> E. x e. R A = D )   &    |- ( F e. S <-> E. y e. R B = F )   &    |- ( G e. S <-> E. z e. R C = G )   &    |- ( A = D -> ( ph <-> ps ) )   &    |- ( B = F -> ( ps <-> ch ) )   &    |- ( C = G -> ( ch <-> th ) )   &    |- ( ( x e. R /\ y e. R /\ z e. R ) -> ph )   =>    |- ( ( D e. S /\ F e. S /\ G e. S ) -> th )
Theorem	cgsexg 3238*	Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
|- ( x = A -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )
Theorem	cgsex2g 3239*	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
|- ( ( x = A /\ y = B ) -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ch /\ ph ) <-> ps ) )
Theorem	cgsex4g 3240*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )   &    |- ( ch -> ( ph <-> ps ) )   =>    |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )
Theorem	ceqsex 3241*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x ( x = A /\ ph ) <-> ps )
Theorem	ceqsexv 3242*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x ( x = A /\ ph ) <-> ps )
Theorem	ceqsexv2d 3243*	Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ps   =>    |- E. x ph
Theorem	ceqsex2 3244*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
|- F/ x ps   &    |- F/ y ch   &    |- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )
Theorem	ceqsex2v 3245*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )
Theorem	ceqsex3v 3246*	Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   =>    |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )
Theorem	ceqsex4v 3247*	Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   =>    |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta )
Theorem	ceqsex6v 3248*	Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   &    |- ( v = E -> ( ta <-> et ) )   &    |- ( u = F -> ( et <-> ze ) )   =>    |- ( E. x E. y E. z E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ze )
Theorem	ceqsex8v 3249*	Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   &    |- G e. _V   &    |- H e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( w = D -> ( th <-> ta ) )   &    |- ( v = E -> ( ta <-> et ) )   &    |- ( u = F -> ( et <-> ze ) )   &    |- ( t = G -> ( ze <-> si ) )   &    |- ( s = H -> ( si <-> rh ) )   =>    |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh )
Theorem	gencbvex 3250*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th <-> E. x ( ch /\ A = y ) )   =>    |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )
Theorem	gencbvex2 3251*	Restatement of gencbvex 3250 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th -> E. x ( ch /\ A = y ) )   =>    |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )
Theorem	gencbval 3252*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
|- A e. _V   &    |- ( A = y -> ( ph <-> ps ) )   &    |- ( A = y -> ( ch <-> th ) )   &    |- ( th <-> E. x ( ch /\ A = y ) )   =>    |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) )
Theorem	sbhypf 3253*	Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3552. (Contributed by Raph Levien, 10-Apr-2004.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( y = A -> ( [ y / x ] ph <-> ps ) )
Theorem	vtoclgft 3254	Closed theorem form of vtoclgf 3264. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.)
|- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )
Theorem	vtoclgftOLD 3255	Obsolete proof of vtoclgft 3254 as of 11-Aug-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )
Theorem	vtocldf 3256	Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ps )   &    |- F/ x ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ch )   =>    |- ( ph -> ch )
Theorem	vtocld 3257*	Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ph -> ps )   =>    |- ( ph -> ch )
Theorem	vtoclf 3258*	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2262. (Contributed by NM, 30-Aug-1993.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl 3259*	Implicit substitution of a class for a setvar variable. See also vtoclALT 3260. (Contributed by NM, 30-Aug-1993.) Removed dependency on ax-10 2019. (Revised by BJ, 29-Nov-2020.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtoclALT 3260*	Alternate proof of vtocl 3259. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl2 3261*	Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtocl3 3262*	Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	vtoclb 3263*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( ph <-> ps )   =>    |- ( ch <-> th )
Theorem	vtoclgf 3264	Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg1f 3265*	Version of vtoclgf 3264 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 2034 and ax-13 2246. (Contributed by BJ, 1-May-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclg 3266*	Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ph   =>    |- ( A e. V -> ps )
Theorem	vtoclbg 3267*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
|- ( x = A -> ( ph <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( ph <-> ps )   =>    |- ( A e. V -> ( ch <-> th ) )
Theorem	vtocl2gf 3268	Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W ) -> ch )
Theorem	vtocl3gf 3269	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   &    |- F/_ y B   &    |- F/_ z B   &    |- F/_ z C   &    |- F/ x ps   &    |- F/ y ch   &    |- F/ z th   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> th )
Theorem	vtocl2g 3270*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ph   =>    |- ( ( A e. V /\ B e. W ) -> ch )
Theorem	vtoclgaf 3271*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x e. B -> ph )   =>    |- ( A e. B -> ps )
Theorem	vtoclga 3272*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x e. B -> ph )   =>    |- ( A e. B -> ps )
Theorem	vtocl2gaf 3273*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( ( x e. C /\ y e. D ) -> ph )   =>    |- ( ( A e. C /\ B e. D ) -> ch )
Theorem	vtocl2ga 3274*	Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( ( x e. C /\ y e. D ) -> ph )   =>    |- ( ( A e. C /\ B e. D ) -> ch )
Theorem	vtocl3gaf 3275*	Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   &    |- F/_ y B   &    |- F/_ z B   &    |- F/_ z C   &    |- F/ x ps   &    |- F/ y ch   &    |- F/ z th   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph )   =>    |- ( ( A e. R /\ B e. S /\ C e. T ) -> th )
Theorem	vtocl3ga 3276*	Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )   =>    |- ( ( A e. D /\ B e. R /\ C e. S ) -> th )
Theorem	vtocl4g 3277*	Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> rh ) )   &    |- ( w = D -> ( rh <-> th ) )   &    |- ph   =>    |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )
Theorem	vtocl4ga 3278*	Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> rh ) )   &    |- ( w = D -> ( rh <-> th ) )   &    |- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )   =>    |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )
Theorem	vtocleg 3279*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)
|- ( x = A -> ph )   =>    |- ( A e. V -> ph )
Theorem	vtoclegft 3280*	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3281.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
|- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )
Theorem	vtoclef 3281*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
|- F/ x ph   &    |- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtocle 3282*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
|- A e. _V   &    |- ( x = A -> ph )   =>    |- ph
Theorem	vtoclri 3283*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
|- ( x = A -> ( ph <-> ps ) )   &    |- A. x e. B ph   =>    |- ( A e. B -> ps )
Theorem	spcimgft 3284	A closed version of spcimgf 3286. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )
Theorem	spcgft 3285	A closed version of spcgf 3288. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
|- F/ x ps   &    |- F/_ x A   =>    |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )
Theorem	spcimgf 3286	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcimegf 3287	Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spcgf 3288	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcegf 3289	Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spcimdv 3290*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcdv 3291*	Rule of specialization, using implicit substitution. Analogous to rspcdv 3312. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	spcimedv 3292*	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 ps ) )
Theorem	spcgv 3293*	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x ph -> ps ) )
Theorem	spcegv 3294*	Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( ps -> E. x ph ) )
Theorem	spc2egv 3295*	Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) )
Theorem	spc2gv 3296*	Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) )
Theorem	spc3egv 3297*	Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )
Theorem	spc3gv 3298*	Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x A. y A. z ph -> ps ) )
Theorem	spcv 3299*	Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spcev 3300*	Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( ps -> E. x ph )