Theorem cbv1h 1978

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

Hypotheses

Ref	Expression
cbv1h.1	|- (ph -> (ps -> A.yps))
cbv1h.2	|- (ph -> (ch -> A.xch))
cbv1h.3	|- (ph -> (x = y -> (ps -> ch)))

Assertion

Ref	Expression
cbv1h	|- (A.xA.yph -> (A.xps -> A.ych))

Proof of Theorem cbv1h

Step	Hyp	Ref	Expression
1		cbv1h.1	. . . . 5 |- (ph -> (ps -> A.yps))
2	1	sps 1754	. . . 4 |- (A.yph -> (ps -> A.yps))
3	2	al2imi 1561	. . 3 |- (A.xA.yph -> (A.xps -> A.xA.yps))
4		ax-7 1734	. . 3 |- (A.xA.yps -> A.yA.xps)
5	3, 4	syl6 29	. 2 |- (A.xA.yph -> (A.xps -> A.yA.xps))
6		cbv1h.3	. . . . . . . 8 |- (ph -> (x = y -> (ps -> ch)))
7	6	com23 72	. . . . . . 7 |- (ph -> (ps -> (x = y -> ch)))
8		cbv1h.2	. . . . . . 7 |- (ph -> (ch -> A.xch))
9	7, 8	syl6d 64	. . . . . 6 |- (ph -> (ps -> (x = y -> A.xch)))
10	9	al2imi 1561	. . . . 5 |- (A.xph -> (A.xps -> A.x(x = y -> A.xch)))
11		ax9o 1950	. . . . 5 |- (A.x(x = y -> A.xch) -> ch)
12	10, 11	syl6 29	. . . 4 |- (A.xph -> (A.xps -> ch))
13	12	al2imi 1561	. . 3 |- (A.yA.xph -> (A.yA.xps -> A.ych))
14	13	a7s 1735	. 2 |- (A.xA.yph -> (A.yA.xps -> A.ych))
15	5, 14	syld 40	1 |- (A.xA.yph -> (A.xps -> A.ych))

Syntax hints:   -> wi 4  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:  cbv1  1979  cbv2h  1980  cbv3h  1983