Theorem cbval 1678

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 1453	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1453	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1677	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   F/wnf 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by:  sb8  1778  cbval2  1838  sb8eu  1955  abbi  2193  cleqf  2243  cbvralf  2572  ralab2  2757  cbvralcsf  2965  dfss2f  2991  elintab  3655  cbviota  4902  sb8iota  4904  dffun6f  4945  dffun4f  4948  mptfvex  5288  findcard2  6423  findcard2s  6424