Theorem cbv2h 1980

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbv2h.1	|- (ph -> (ps -> A.yps))
cbv2h.2	|- (ph -> (ch -> A.xch))
cbv2h.3	|- (ph -> (x = y -> (ps <-> ch)))

Assertion

Ref	Expression
cbv2h	|- (A.xA.yph -> (A.xps <-> A.ych))

Proof of Theorem cbv2h

Step	Hyp	Ref	Expression
1		cbv2h.1	. . 3 |- (ph -> (ps -> A.yps))
2		cbv2h.2	. . 3 |- (ph -> (ch -> A.xch))
3		cbv2h.3	. . . 4 |- (ph -> (x = y -> (ps <-> ch)))
4		bi1 178	. . . 4 |- ((ps <-> ch) -> (ps -> ch))
5	3, 4	syl6 29	. . 3 |- (ph -> (x = y -> (ps -> ch)))
6	1, 2, 5	cbv1h 1978	. 2 |- (A.xA.yph -> (A.xps -> A.ych))
7		equcomi 1679	. . . . 5 |- (y = x -> x = y)
8		bi2 189	. . . . 5 |- ((ps <-> ch) -> (ch -> ps))
9	7, 3, 8	syl56 30	. . . 4 |- (ph -> (y = x -> (ch -> ps)))
10	2, 1, 9	cbv1h 1978	. . 3 |- (A.yA.xph -> (A.ych -> A.xps))
11	10	a7s 1735	. 2 |- (A.xA.yph -> (A.ych -> A.xps))
12	6, 11	impbid 183	1 |- (A.xA.yph -> (A.xps <-> A.ych))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540

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:  cbv2  1981  eujustALT  2207