Theorem cbvalv 1917

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 1526	. 2 |- ( ph -> A. y ph )
2		ax-17 1526	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1753	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:  nfcjust  2307  cdeqal1  2953  dfss4st  3368  zfpow  4175  tfisi  4586  acexmid  5873  tfrlem3-2d  6312  tfrlemi1  6332  tfrexlem  6334  tfr1onlemaccex  6348  tfrcllemaccex  6361  findcard  6887  fisseneq  6930  genprndl  7519  genprndu  7520  zfz1iso  10820