Theorem vtoclf 2876

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1810. (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 2830	. . . 4 |- E. x x = A
4		vtoclf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 144	. . . 4 |- ( x = A -> ( ph -> ps ) )
6	3, 5	eximii 1655	. . 3 |- E. x ( ph -> ps )
7	1, 6	19.36i 1724	. 2 |- ( A. x ph -> ps )
8		vtoclf.4	. 2 |- ph
9	7, 8	mpg 1504	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   F/wnf 1513   e. wcel 2209   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  vtocl  2877