Theorem cbval 1778

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 1543	. 2 |- ( ph -> A. y ph )
3		cbval.2	. . 3 |- F/ x ps
4	3	nfri 1543	. 2 |- ( ps -> A. x ps )
5		cbval.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvalh 1777	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   F/wnf 1484

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  sb8  1880  cbval2  1946  sb8eu  2068  abbi  2320  cleqf  2374  cbvralf  2731  ralab2  2941  cbvralcsf  3160  dfss2f  3188  elintab  3902  cbviota  5246  sb8iota  5248  dffun6f  5293  dffun4f  5296  mptfvex  5678  findcard2  7001  findcard2s  7002