Theorem cbv2 1163

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

Hypotheses

Ref	Expression
cbv2.1	|- (ph -> (ps -> A.yps))
cbv2.2	|- (ph -> (ch -> A.xch))
cbv2.3	|- (ph -> (x = y -> (ps <-> ch)))

Assertion

Ref	Expression
cbv2	|- (A.xA.yph -> (A.xps <-> A.ych))

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.1	. . 3 |- (ph -> (ps -> A.yps))
2		cbv2.2	. . 3 |- (ph -> (ch -> A.xch))
3		cbv2.3	. . . 4 |- (ph -> (x = y -> (ps <-> ch)))
4		bi1 148	. . . 4 |- ((ps <-> ch) -> (ps -> ch))
5	3, 4	syl6 22	. . 3 |- (ph -> (x = y -> (ps -> ch)))
6	1, 2, 5	cbv1 1162	. 2 |- (A.xA.yph -> (A.xps -> A.ych))
7		bi2 149	. . . . . 6 |- ((ps <-> ch) -> (ch -> ps))
8	3, 7	syl6 22	. . . . 5 |- (ph -> (x = y -> (ch -> ps)))
9		equcomi 1128	. . . . 5 |- (y = x -> x = y)
10	8, 9	syl5 21	. . . 4 |- (ph -> (y = x -> (ch -> ps)))
11	2, 1, 10	cbv1 1162	. . 3 |- (A.yA.xph -> (A.ych -> A.xps))
12	11	a7s 991	. 2 |- (A.xA.yph -> (A.ych -> A.xps))
13	6, 12	impbid 516	1 |- (A.xA.yph -> (A.xps <-> A.ych))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956

This theorem is referenced by:  cbval 1165

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-an 225

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