Theorem cbval 1684

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 1457	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1457	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1683	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   F/wnf 1394

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  sb8  1784  cbval2  1844  sb8eu  1961  abbi  2201  cleqf  2252  cbvralf  2584  ralab2  2777  cbvralcsf  2988  dfss2f  3014  elintab  3694  cbviota  4972  sb8iota  4974  dffun6f  5015  dffun4f  5018  mptfvex  5372  findcard2  6585  findcard2s  6586