Theorem cbv1 1158

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbv1.1	|- (ph -> (ps -> A.yps))
cbv1.2	|- (ph -> (ch -> A.xch))
cbv1.3	|- (ph -> (x = y -> (ps -> ch)))

Assertion

Ref	Expression
cbv1	|- (A.xA.yph -> (A.xps -> A.ych))

Proof of Theorem cbv1

Step	Hyp	Ref	Expression
1		cbv1.1	. . . . 5 |- (ph -> (ps -> A.yps))
2	1	a4s 981	. . . 4 |- (A.yph -> (ps -> A.yps))
3	2	19.20ii 992	. . 3 |- (A.xA.yph -> (A.xps -> A.xA.yps))
4		ax-7 959	. . 3 |- (A.xA.yps -> A.yA.xps)
5	3, 4	syl6 22	. 2 |- (A.xA.yph -> (A.xps -> A.yA.xps))
6		cbv1.3	. . . . . . . 8 |- (ph -> (x = y -> (ps -> ch)))
7	6	com23 32	. . . . . . 7 |- (ph -> (ps -> (x = y -> ch)))
8		cbv1.2	. . . . . . 7 |- (ph -> (ch -> A.xch))
9	7, 8	syl6d 56	. . . . . 6 |- (ph -> (ps -> (x = y -> A.xch)))
10	9	19.20ii 992	. . . . 5 |- (A.xph -> (A.xps -> A.x(x = y -> A.xch)))
11		ax-9o 1119	. . . . 5 |- (A.x(x = y -> A.xch) -> ch)
12	10, 11	syl6 22	. . . 4 |- (A.xph -> (A.xps -> ch))
13	12	19.20ii 992	. . 3 |- (A.yA.xph -> (A.yA.xps -> A.ych))
14	13	a7s 988	. 2 |- (A.xA.yph -> (A.yA.xps -> A.ych))
15	5, 14	syld 27	1 |- (A.xA.yph -> (A.xps -> A.ych))

Syntax hints:   -> wi 3  A.wal 951   = wceq 953

This theorem is referenced by:  cbv2 1159  cbv3 1160

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-mp 7  ax-7 959  ax-gen 960  ax-4 970  ax-5o 972  ax-9o 1119

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