Theorem cbvalv 1905

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 1514	. 2 |- ( ph -> A. y ph )
2		ax-17 1514	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1741	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  nfcjust  2296  cdeqal1  2942  dfss4st  3355  zfpow  4154  tfisi  4564  acexmid  5841  tfrlem3-2d  6280  tfrlemi1  6300  tfrexlem  6302  tfr1onlemaccex  6316  tfrcllemaccex  6329  findcard  6854  fisseneq  6897  genprndl  7462  genprndu  7463  zfz1iso  10754