Theorem cbvalv 1942

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 1550	. 2 |- ( ph -> A. y ph )
2		ax-17 1550	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1777	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  nfcjust  2338  cdeqal1  2996  dfss4st  3414  zfpow  4235  tfisi  4653  acexmid  5966  tfrlem3-2d  6421  tfrlemi1  6441  tfrexlem  6443  tfr1onlemaccex  6457  tfrcllemaccex  6470  findcard  7011  fisseneq  7057  genprndl  7669  genprndu  7670  zfz1iso  11023