Theorem cbvexd 1316

Description: Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1347.

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 groups:   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 1105	. . . 4 |- (ph -> (-. ps -> A.y -. ps))
4		cbvald.3	. . . . 5 |- (ph -> (x = y -> (ps <-> ch)))
5		pm4.11 520	. . . . 5 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
6	4, 5	syl6ib 212	. . . 4 |- (ph -> (x = y -> (-. ps <-> -. ch)))
7	1, 3, 6	cbvald 1315	. . 3 |- (ph -> (A.x -. ps <-> A.y -. ch))
8	7	negbid 609	. 2 |- (ph -> (-. A.x -. ps <-> -. A.y -. ch))
9		df-ex 978	. 2 |- (E.xps <-> -. A.x -. ps)
10		df-ex 978	. 2 |- (E.ych <-> -. A.y -. ch)
11	8, 9, 10	3bitr4g 553	1 |- (ph -> (E.xps <-> E.ych))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 951   = wceq 953  E.wex 977

This theorem is referenced by:  dfid3 2826  axrepndlem2 4917  axunnd 4920  axpowndlem2 4922  axpownd 4925  axregndlem2 4927  axinfndlem1 4929  axacndlem4 4934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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