Theorem wl-eudf 33354

Description: Version of df-eu 2474 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)

Hypotheses

Ref	Expression
wl-eudf.1	|- F/ x ph
wl-eudf.2	|- F/ y ph
wl-eudf.3	|- ( ph -> -. A. x x = y )
wl-eudf.4	|- ( ph -> F/ y ps )

Assertion

Ref	Expression
wl-eudf	|- ( ph -> ( E! x ps <-> E. y A. x ( ps <-> x = y ) ) )

Proof of Theorem wl-eudf

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2474	. 2 |- ( E! x ps <-> E. u A. x ( ps <-> x = u ) )
2		wl-eudf.2	. . 3 |- F/ y ph
3		wl-eudf.1	. . . 4 |- F/ x ph
4		wl-eudf.4	. . . . 5 |- ( ph -> F/ y ps )
5		wl-eudf.3	. . . . . 6 |- ( ph -> -. A. x x = y )
6		nfeqf1 2299	. . . . . . 7 |- ( -. A. y y = x -> F/ y x = u )
7	6	naecoms 2313	. . . . . 6 |- ( -. A. x x = y -> F/ y x = u )
8	5, 7	syl 17	. . . . 5 |- ( ph -> F/ y x = u )
9	4, 8	nfbid 1832	. . . 4 |- ( ph -> F/ y ( ps <-> x = u ) )
10	3, 9	nfald 2165	. . 3 |- ( ph -> F/ y A. x ( ps <-> x = u ) )
11		nfnae 2318	. . . . . . 7 |- F/ x -. A. x x = y
12		nfeqf2 2297	. . . . . . 7 |- ( -. A. x x = y -> F/ x u = y )
13	11, 12	nfan1 2068	. . . . . 6 |- F/ x ( -. A. x x = y /\ u = y )
14		equequ2 1953	. . . . . . . 8 |- ( u = y -> ( x = u <-> x = y ) )
15	14	bibi2d 332	. . . . . . 7 |- ( u = y -> ( ( ps <-> x = u ) <-> ( ps <-> x = y ) ) )
16	15	adantl 482	. . . . . 6 |- ( ( -. A. x x = y /\ u = y ) -> ( ( ps <-> x = u ) <-> ( ps <-> x = y ) ) )
17	13, 16	albid 2090	. . . . 5 |- ( ( -. A. x x = y /\ u = y ) -> ( A. x ( ps <-> x = u ) <-> A. x ( ps <-> x = y ) ) )
18	5, 17	sylan 488	. . . 4 |- ( ( ph /\ u = y ) -> ( A. x ( ps <-> x = u ) <-> A. x ( ps <-> x = y ) ) )
19	18	ex 450	. . 3 |- ( ph -> ( u = y -> ( A. x ( ps <-> x = u ) <-> A. x ( ps <-> x = y ) ) ) )
20	2, 10, 19	cbvexd 2278	. 2 |- ( ph -> ( E. u A. x ( ps <-> x = u ) <-> E. y A. x ( ps <-> x = y ) ) )
21	1, 20	syl5bb 272	1 |- ( ph -> ( E! x ps <-> E. y A. x ( ps <-> x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   E!weu 2470

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-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by:  wl-eutf  33355