Theorem cbvald 2008

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
cbvald.1	|- F/yph
cbvald.2	|- (ph -> F/yps)
cbvald.3	|- (ph -> (x = y -> (ps <-> ch)))

Assertion

Ref	Expression
cbvald	|- (ph -> (A.xps <-> A.ych))

Distinct variable groups:   ph,x   ch,x

Allowed substitution hints:   ph(y)   ps(x,y)   ch(y)

Proof of Theorem cbvald

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 |- F/yph
2	1	nfri 1762	. . 3 |- (ph -> A.yph)
3	2	alrimiv 1631	. 2 |- (ph -> A.xA.yph)
4		cbvald.2	. . 3 |- (ph -> F/yps)
5		nfvd 1620	. . 3 |- (ph -> F/xch)
6		cbvald.3	. . 3 |- (ph -> (x = y -> (ps <-> ch)))
7	4, 5, 6	cbv2 1981	. 2 |- (A.xA.yph -> (A.xps <-> A.ych))
8	3, 7	syl 15	1 |- (ph -> (A.xps <-> A.ych))

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:  cbvexd  2009  cbvaldva  2010