Theorem cbvexd 2009

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ph,x   ch,x

Allowed substitution hints:   ph(y)   ps(x,y)   ch(y)

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 |- F/yph
2		cbvald.2	. . . . 5 |- (ph -> F/yps)
3	2	nfnd 1791	. . . 4 |- (ph -> F/y -. ps)
4		cbvald.3	. . . . 5 |- (ph -> (x = y -> (ps <-> ch)))
5		notbi 286	. . . . 5 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
6	4, 5	syl6ib 217	. . . 4 |- (ph -> (x = y -> (-. ps <-> -. ch)))
7	1, 3, 6	cbvald 2008	. . 3 |- (ph -> (A.x -. ps <-> A.y -. ch))
8	7	notbid 285	. 2 |- (ph -> (-. A.x -. ps <-> -. A.y -. ch))
9		df-ex 1542	. 2 |- (E.xps <-> -. A.x -. ps)
10		df-ex 1542	. 2 |- (E.ych <-> -. A.y -. ch)
11	8, 9, 10	3bitr4g 279	1 |- (ph -> (E.xps <-> E.ych))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvexdva  2011  vtoclgft  2906  dfid3  4769