Theorem cbvalh 1753

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
cbvalh.1	|- ( ph -> A. y ph )
cbvalh.2	|- ( ps -> A. x ps )
cbvalh.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvalh	|- ( A. x ph <-> A. y ps )

Proof of Theorem cbvalh

Step	Hyp	Ref	Expression
1		cbvalh.1	. . 3 |- ( ph -> A. y ph )
2		cbvalh.2	. . 3 |- ( ps -> A. x ps )
3		cbvalh.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	biimpd 144	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbv3h 1743	. 2 |- ( A. x ph -> A. y ps )
6	3	equcoms 1708	. . . 4 |- ( y = x -> ( ph <-> ps ) )
7	6	biimprd 158	. . 3 |- ( y = x -> ( ps -> ph ) )
8	2, 1, 7	cbv3h 1743	. 2 |- ( A. y ps -> A. x ph )
9	5, 8	impbii 126	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  cbval  1754  sb8h  1854  cbvalv  1917  sb9v  1978  sb8euh  2049