Theorem cbvexd 2059

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2066. (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 1805	. . . 4 |- ( ph -> F/ y -. ps )
4		cbvald.3	. . . . 5 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5		notbi 287	. . . . 5 |- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
6	4, 5	syl6ib 218	. . . 4 |- ( ph -> ( x = y -> ( -. ps <-> -. ch ) ) )
7	1, 3, 6	cbvald 2058	. . 3 |- ( ph -> ( A. x -. ps <-> A. y -. ch ) )
8	7	notbid 286	. 2 |- ( ph -> ( -. A. x -. ps <-> -. A. y -. ch ) )
9		df-ex 1548	. 2 |- ( E. x ps <-> -. A. x -. ps )
10		df-ex 1548	. 2 |- ( E. y ch <-> -. A. y -. ch )
11	8, 9, 10	3bitr4g 280	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550

This theorem is referenced by:  cbvexdva  2061  vtoclgft  2962  dfid3  4459  axrepndlem2  8424  axunnd  8427  axpowndlem2  8429  axpownd  8432  axregndlem2  8434  axinfndlem1  8436  axacndlem4  8441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551