Theorem cbvalv 1973

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 1579	. 2 |- ( ph -> A. y ph )
2		ax-17 1579	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvalh 1806	1 |- ( A. x ph <-> A. y ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wal 1400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  nfcjust  2380  cdeqal1  3042  dfss4st  3464  zfpow  4314  tfisi  4736  acexmid  6087  tfrlem3-2d  6587  tfrlemi1  6607  tfrexlem  6609  tfr1onlemaccex  6623  tfrcllemaccex  6636  findcard  7196  fisseneq  7246  genprndl  7894  genprndu  7895  zfz1iso  11315