Theorem cbval 1984

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		cbval.2	. . 3 |- F/xps
3		cbval.3	. . . 4 |- (x = y -> (ph <-> ps))
4	3	biimpd 198	. . 3 |- (x = y -> (ph -> ps))
5	1, 2, 4	cbv3 1982	. 2 |- (A.xph -> A.yps)
6	3	biimprd 214	. . . 4 |- (x = y -> (ps -> ph))
7	6	equcoms 1681	. . 3 |- (y = x -> (ps -> ph))
8	2, 1, 7	cbv3 1982	. 2 |- (A.yps -> A.xph)
9	5, 8	impbii 180	1 |- (A.xph <-> A.yps)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvex  1985  cbvalv  2002  cbval2  2004  sb8eu  2222  abbi  2464  cleqf  2514  cbvralf  2830  ralab2  3002  cbvralcsf  3199  dfss2f  3265  elintab  3938  cbviota  4345  sb8iota  4347  dffun6f  5124