Theorem vtoclf 3258

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2262. (Contributed by NM, 30-Aug-1993.)

Hypotheses

Ref	Expression
vtoclf.1	|- F/ x ps
vtoclf.2	|- A e. _V
vtoclf.3	|- ( x = A -> ( ph <-> ps ) )
vtoclf.4	|- ph

Assertion

Ref	Expression
vtoclf	|- ps

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. . 3 |- F/ x ps
2		vtoclf.2	. . . . 5 |- A e. _V
3	2	isseti 3209	. . . 4 |- E. x x = A
4		vtoclf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 219	. . . 4 |- ( x = A -> ( ph -> ps ) )
6	3, 5	eximii 1764	. . 3 |- E. x ( ph -> ps )
7	1, 6	19.36i 2099	. 2 |- ( A. x ph -> ps )
8		vtoclf.4	. 2 |- ph
9	7, 8	mpg 1724	1 |- ps

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  vtoclALT  3260  summolem2a  14446  prodmolem2a  14664  poimirlem24  33433  poimirlem28  33437  monotuz  37506  oddcomabszz  37509  binomcxplemnotnn0  38555  limclner  39883  climinf2mpt  39946  climinfmpt  39947  dvnmptdivc  40153  dvnmul  40158  salpreimagtge  40934  salpreimaltle  40935