Theorem vtoclb 2690

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)

Hypotheses

Ref	Expression
vtoclb.1	|- A e. _V
vtoclb.2	|- ( x = A -> ( ph <-> ch ) )
vtoclb.3	|- ( x = A -> ( ps <-> th ) )
vtoclb.4	|- ( ph <-> ps )

Assertion

Ref	Expression
vtoclb	|- ( ch <-> th )

Distinct variable groups:   x, A   ch, x   th, x

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

Proof of Theorem vtoclb

Step	Hyp	Ref	Expression
1		vtoclb.1	. 2 |- A e. _V
2		vtoclb.2	. . 3 |- ( x = A -> ( ph <-> ch ) )
3		vtoclb.3	. . 3 |- ( x = A -> ( ps <-> th ) )
4	2, 3	bibi12d 234	. 2 |- ( x = A -> ( ( ph <-> ps ) <-> ( ch <-> th ) ) )
5		vtoclb.4	. 2 |- ( ph <-> ps )
6	1, 4, 5	vtocl 2687	1 |- ( ch <-> th )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1296   e. wcel 1445   _Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-v 2635

This theorem is referenced by:  alexeq  2757  sbss  3410