Theorem vtoclb 2611

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 224	. 2 |- ( x = A -> ( ( ph <-> ps ) <-> ( ch <-> th ) ) )
5		vtoclb.4	. 2 |- ( ph <-> ps )
6	1, 4, 5	vtocl 2608	1 |- ( ch <-> th )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   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:  alexeq  2670  sbss  3329