Theorem cbval 1752

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 1517	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1517	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1751	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   F/wnf 1458

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  sb8  1854  cbval2  1919  sb8eu  2037  abbi  2289  cleqf  2342  cbvralf  2694  ralab2  2899  cbvralcsf  3117  dfss2f  3144  elintab  3851  cbviota  5175  sb8iota  5177  dffun6f  5221  dffun4f  5224  mptfvex  5593  findcard2  6879  findcard2s  6880