Theorem cbvexd 2033

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2085. (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 1910	. . . 4 |- ( ph -> F/ y -. ps )
4		cbvald.3	. . . . 5 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5		notbi 293	. . . . 5 |- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
6	4, 5	syl6ib 226	. . . 4 |- ( ph -> ( x = y -> ( -. ps <-> -. ch ) ) )
7	1, 3, 6	cbvald 2032	. . 3 |- ( ph -> ( A. x -. ps <-> A. y -. ch ) )
8	7	notbid 292	. 2 |- ( ph -> ( -. A. x -. ps <-> -. A. y -. ch ) )
9		df-ex 1621	. 2 |- ( E. x ps <-> -. A. x -. ps )
10		df-ex 1621	. 2 |- ( E. y ch <-> -. A. y -. ch )
11	8, 9, 10	3bitr4g 288	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   A.wal 1397   E.wex 1620   F/wnf 1624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625

This theorem is referenced by:  cbvexdva  2039  vtoclgft  3082  dfid3  4710  axrepndlem2  8881  axunnd  8884  axpowndlem2  8886  axpownd  8889  axregndlem2  8891  axinfndlem1  8894  axacndlem4  8899  wl-mo2dnae  30184  wl-mo2t  30185