Theorem vtoclb 2783

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 2780	1 |- ( ch <-> th )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  alexeq  2852  sbss  3517