Theorem vtoclf 2607

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1640. (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 2563	. . . 4 |- E. x x = A
4		vtoclf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 132	. . . 4 |- ( x = A -> ( ph -> ps ) )
6	3, 5	eximii 1493	. . 3 |- E. x ( ph -> ps )
7	1, 6	19.36i 1562	. 2 |- ( A. x ph -> ps )
8		vtoclf.4	. 2 |- ph
9	7, 8	mpg 1340	1 |- ps

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  vtocl  2608