Theorem vtoclb 3263

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 335	. 2 |- ( x = A -> ( ( ph <-> ps ) <-> ( ch <-> th ) ) )
5		vtoclb.4	. 2 |- ( ph <-> ps )
6	1, 4, 5	vtocl 3259	1 |- ( ch <-> th )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  sbss  4084  bnj609  30987