Theorem cbval 1765

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
cbval.1	|- F/ y ph
cbval.2	|- F/ x ps
cbval.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbval	|- ( A. x ph <-> A. y ps )

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . 3 |- F/ y ph
2	1	nfri 1530	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1530	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1764	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   F/wnf 1471

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  sb8  1867  cbval2  1933  sb8eu  2055  abbi  2307  cleqf  2361  cbvralf  2718  ralab2  2925  cbvralcsf  3144  dfss2f  3171  elintab  3882  cbviota  5221  sb8iota  5223  dffun6f  5268  dffun4f  5271  mptfvex  5644  findcard2  6947  findcard2s  6948