Theorem cbvalv 1964

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 1572	. 2 |- ( ph -> A. y ph )
2		ax-17 1572	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1799	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  nfcjust  2360  cdeqal1  3019  dfss4st  3437  zfpow  4259  tfisi  4679  acexmid  6000  tfrlem3-2d  6458  tfrlemi1  6478  tfrexlem  6480  tfr1onlemaccex  6494  tfrcllemaccex  6507  findcard  7050  fisseneq  7096  genprndl  7708  genprndu  7709  zfz1iso  11063