Theorem cbvalh 1636

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 132	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbv3h 1631	. 2 |- ( A. x ph -> A. y ps )
6	3	equcoms 1594	. . . 4 |- ( y = x -> ( ph <-> ps ) )
7	6	biimprd 147	. . 3 |- ( y = x -> ( ps -> ph ) )
8	2, 1, 7	cbv3h 1631	. 2 |- ( A. y ps -> A. x ph )
9	5, 8	impbii 117	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241

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-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbval  1637  sb8h  1734  cbvalv  1794  sb9v  1854  sb8euh  1923