Theorem cbval 1634

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/yph
cbval.2	|- F/xps
cbval.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbval	|- (A.xph <-> A.yps)

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . 3 |- F/yph
2	1	nfri 1409	. 2 |- (ph -> A.yph)
3		cbval.2	. . 3 |- F/xps
4	3	nfri 1409	. 2 |- (ps -> A.xps)
5		cbval.3	. 2 |- (x = y -> (ph <-> ps))
6	2, 4, 5	cbvalh 1633	1 |- (A.xph <-> A.yps)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   F/wnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  sb8  1733  cbval2  1793  sb8eu  1910  abbi  2148  cleqf  2198  cbvralf  2521  ralab2  2699  cbvralcsf  2902  dfss2f  2930  elintab  3617  cbviota  4815  sb8iota  4817  dffun6f  4858  dffun4f  4861  mptfvex  5199