Theorem cbvalOLD 1528

Description: Rule used to change bound variables, using implicit substitition.

Hypotheses

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

Assertion

Ref	Expression
cbvalOLD	|- (A.xph <-> A.yps)

Proof of Theorem cbvalOLD

Step	Hyp	Ref	Expression
1		cbval.1	. . . 4 |- (ph -> A.yph)
2	1	imim2i 11	. . 3 |- ((ph -> ph) -> (ph -> A.yph))
3		cbval.2	. . . 4 |- (ps -> A.xps)
4	3	a1i 8	. . 3 |- ((ph -> ph) -> (ps -> A.xps))
5		cbval.3	. . . 4 |- (x = y -> (ph <-> ps))
6	5	a1i 8	. . 3 |- ((ph -> ph) -> (x = y -> (ph <-> ps)))
7	2, 4, 6	cbv2 1524	. 2 |- (A.xA.y(ph -> ph) -> (A.xph <-> A.yps))
8		id 73	. . 3 |- (ph -> ph)
9	8	ax-gen 1305	. 2 |- A.y(ph -> ph)
10	7, 9	mpg 1332	1 |- (A.xph <-> A.yps)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-an 242

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