Theorem cbv3 847

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

Hypotheses

Ref	Expression
cbv3.1	|- (ph -> A.yph)
cbv3.2	|- (ps -> A.xps)
cbv3.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbv3	|- (A.xph -> A.yps)

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . . 4 |- (ph -> A.yph)
2	1	syl3 18	. . 3 |- ((ph -> ph) -> (ph -> A.yph))
3		cbv3.2	. . . 4 |- (ps -> A.xps)
4	3	a1i 7	. . 3 |- ((ph -> ph) -> (ps -> A.xps))
5		cbv3.3	. . . 4 |- (x = y -> (ph -> ps))
6	5	a1i 7	. . 3 |- ((ph -> ph) -> (x = y -> (ph -> ps)))
7	2, 4, 6	cbv1 845	. 2 |- (A.xA.y(ph -> ph) -> (A.xph -> A.yps))
8		id 9	. . 3 |- (ph -> ph)
9	8	ax-gen 677	. 2 |- A.y(ph -> ph)
10	7, 9	mpg 684	1 |- (A.xph -> A.yps)

Syntax hints:   -> wi 2  A.wal 672   = weq 797

This theorem is referenced by:  mo 1020

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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