Theorem cbval 1148

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

Hypotheses

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

Assertion

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

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . . 4 |- (ph -> A.yph)
2	1	imim2i 17	. . 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 1146	. 2 |- (A.xA.y(ph -> ph) -> (A.xph <-> A.yps))
8		id 59	. . 3 |- (ph -> ph)
9	8	ax-gen 955	. 2 |- A.y(ph -> ph)
10	7, 9	mpg 962	1 |- (A.xph <-> A.yps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950   = wceq 1099

This theorem is referenced by:  cbvex 1149  cbvalv 1296  cbval2 1298  cbvald 1302  cleqf 1536  cbvralf 1772  dfss2f 2031  elintab 2512  ssintab 2518  dffunmof 3471  aceq1 4653  nnwof 6342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-12 1104

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

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