Theorem cbvexd 978

Description: Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with ddelim 1000.

Hypotheses

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

Assertion

Ref	Expression
cbvexd	|- (ph -> (E.xps <-> E.ych))

Distinct variable group(s):   ph,x   ch,x

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 |- (ph -> A.yph)
2		cbvald.2	. . . . 5 |- (ph -> (ps -> A.yps))
3	1, 2	hbnd 786	. . . 4 |- (ph -> (-. ps -> A.y -. ps))
4		cbvald.3	. . . . 5 |- (ph -> (x = y -> (ps <-> ch)))
5		pm4.11 400	. . . . 5 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
6	4, 5	syl6ib 185	. . . 4 |- (ph -> (x = y -> (-. ps <-> -. ch)))
7	1, 3, 6	cbvald 977	. . 3 |- (ph -> (A.x -. ps <-> A.y -. ch))
8	7	negbid 463	. 2 |- (ph -> (-. A.x -. ps <-> -. A.y -. ch))
9		df-ex 679	. 2 |- (E.xps <-> -. A.x -. ps)
10		df-ex 679	. 2 |- (E.ych <-> -. A.y -. ch)
11	8, 9, 10	3bitr4g 428	1 |- (ph -> (E.xps <-> E.ych))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  axrepndlem2 3739  axunnd 3742  axpowndlem2 3744  axpownd 3747  axregndlem2 3749  axinfndlem1 3751  axacndlem4 3756

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-8 798  ax-9 799  ax-12 802  ax-17 925

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

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