Theorem cbvexd 2278

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2337. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbvexd	|- ( ph -> ( E. x ps <-> E. y ch ) )

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/ y ph
2		cbvald.2	. . . . 5 |- ( ph -> F/ y ps )
3	2	nfnd 1785	. . . 4 |- ( ph -> F/ y -. ps )
4		cbvald.3	. . . . 5 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5		notbi 309	. . . . 5 |- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
6	4, 5	syl6ib 241	. . . 4 |- ( ph -> ( x = y -> ( -. ps <-> -. ch ) ) )
7	1, 3, 6	cbvald 2277	. . 3 |- ( ph -> ( A. x -. ps <-> A. y -. ch ) )
8	7	notbid 308	. 2 |- ( ph -> ( -. A. x -. ps <-> -. A. y -. ch ) )
9		df-ex 1705	. 2 |- ( E. x ps <-> -. A. x -. ps )
10		df-ex 1705	. 2 |- ( E. y ch <-> -. A. y -. ch )
11	8, 9, 10	3bitr4g 303	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbvexdvaOLD  2284  vtoclgft  3254  vtoclgftOLD  3255  dfid3  5025  axrepndlem2  9415  axunnd  9418  axpowndlem2  9420  axpownd  9423  axregndlem2  9425  axinfndlem1  9427  axacndlem4  9432  wl-mo2df  33352  wl-eudf  33354  wl-mo2t  33357