Theorem cbval 1695

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 1467	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1467	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1694	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1297   F/wnf 1404

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-nf 1405

This theorem is referenced by:  sb8  1795  cbval2  1856  sb8eu  1973  abbi  2213  cleqf  2264  cbvralf  2606  ralab2  2801  cbvralcsf  3012  dfss2f  3038  elintab  3729  cbviota  5029  sb8iota  5031  dffun6f  5072  dffun4f  5075  mptfvex  5438  findcard2  6712  findcard2s  6713