Theorem cbvalv 1910

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 1519	. 2 |- ( ph -> A. y ph )
2		ax-17 1519	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1746	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  nfcjust  2300  cdeqal1  2946  dfss4st  3360  zfpow  4161  tfisi  4571  acexmid  5852  tfrlem3-2d  6291  tfrlemi1  6311  tfrexlem  6313  tfr1onlemaccex  6327  tfrcllemaccex  6340  findcard  6866  fisseneq  6909  genprndl  7483  genprndu  7484  zfz1iso  10776