Theorem reupick 3727

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reupick	|- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) -> ( x e. A <-> x e. B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 3426	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	ad2antrr 732	. 2 |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) -> ( x e. A -> x e. B ) )
3		df-rex 2743	. . . . . 6 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-reu 2744	. . . . . 6 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
5	3, 4	anbi12i 703	. . . . 5 |- ( ( E. x e. A ph /\ E! x e. B ph ) <-> ( E. x ( x e. A /\ ph ) /\ E! x ( x e. B /\ ph ) ) )
6	1	ancrd 557	. . . . . . . . . . 11 |- ( A C_ B -> ( x e. A -> ( x e. B /\ x e. A ) ) )
7	6	anim1d 568	. . . . . . . . . 10 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( ( x e. B /\ x e. A ) /\ ph ) ) )
8		an32 807	. . . . . . . . . 10 |- ( ( ( x e. B /\ x e. A ) /\ ph ) <-> ( ( x e. B /\ ph ) /\ x e. A ) )
9	7, 8	syl6ib 230	. . . . . . . . 9 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( ( x e. B /\ ph ) /\ x e. A ) ) )
10	9	eximdv 1764	. . . . . . . 8 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> E. x ( ( x e. B /\ ph ) /\ x e. A ) ) )
11		eupick 2365	. . . . . . . . 9 |- ( ( E! x ( x e. B /\ ph ) /\ E. x ( ( x e. B /\ ph ) /\ x e. A ) ) -> ( ( x e. B /\ ph ) -> x e. A ) )
12	11	ex 436	. . . . . . . 8 |- ( E! x ( x e. B /\ ph ) -> ( E. x ( ( x e. B /\ ph ) /\ x e. A ) -> ( ( x e. B /\ ph ) -> x e. A ) ) )
13	10, 12	syl9 73	. . . . . . 7 |- ( A C_ B -> ( E! x ( x e. B /\ ph ) -> ( E. x ( x e. A /\ ph ) -> ( ( x e. B /\ ph ) -> x e. A ) ) ) )
14	13	com23 81	. . . . . 6 |- ( A C_ B -> ( E. x ( x e. A /\ ph ) -> ( E! x ( x e. B /\ ph ) -> ( ( x e. B /\ ph ) -> x e. A ) ) ) )
15	14	imp32 435	. . . . 5 |- ( ( A C_ B /\ ( E. x ( x e. A /\ ph ) /\ E! x ( x e. B /\ ph ) ) ) -> ( ( x e. B /\ ph ) -> x e. A ) )
16	5, 15	sylan2b 478	. . . 4 |- ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> ( ( x e. B /\ ph ) -> x e. A ) )
17	16	expcomd 440	. . 3 |- ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> ( ph -> ( x e. B -> x e. A ) ) )
18	17	imp 431	. 2 |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) -> ( x e. B -> x e. A ) )
19	2, 18	impbid 194	1 |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) -> ( x e. A <-> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   E.wex 1663   e. wcel 1887   E!weu 2299   E.wrex 2738   E!wreu 2739   C_ wss 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-rex 2743  df-reu 2744  df-in 3411  df-ss 3418

This theorem is referenced by: (None)