Theorem cbvalv 1897

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
cbvalv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvalv

Step	Hyp	Ref	Expression
1		ax-17 1506	. 2 |- ( ph -> A. y ph )
2		ax-17 1506	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1733	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1333

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441

This theorem is referenced by:  nfcjust  2287  cdeqal1  2928  dfss4st  3340  zfpow  4136  tfisi  4546  acexmid  5823  tfrlem3-2d  6259  tfrlemi1  6279  tfrexlem  6281  tfr1onlemaccex  6295  tfrcllemaccex  6308  findcard  6833  fisseneq  6876  genprndl  7441  genprndu  7442  zfz1iso  10712