Theorem cbvalv 1890

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 1507	. 2 |- ( ph -> A. y ph )
2		ax-17 1507	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1727	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438

This theorem is referenced by:  cbvalvw  1892  nfcjust  2270  cdeqal1  2904  dfss4st  3314  zfpow  4107  tfisi  4509  acexmid  5781  tfrlem3-2d  6217  tfrlemi1  6237  tfrexlem  6239  tfr1onlemaccex  6253  tfrcllemaccex  6266  findcard  6790  fisseneq  6828  genprndl  7353  genprndu  7354  zfz1iso  10616