Theorem reupick 3540

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 3268	. . 3 |- (A C_ B -> (x e. A -> x e. B))
2	1	ad2antrr 706	. 2 |- (((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A -> x e. B))
3		df-rex 2621	. . . . . 6 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-reu 2622	. . . . . 6 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
5	3, 4	anbi12i 678	. . . . 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 537	. . . . . . . . . . 11 |- (A C_ B -> (x e. A -> (x e. B /\ x e. A)))
7	6	anim1d 547	. . . . . . . . . 10 |- (A C_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
8		an32 773	. . . . . . . . . 10 |- (((x e. B /\ x e. A) /\ ph) <-> ((x e. B /\ ph) /\ x e. A))
9	7, 8	syl6ib 217	. . . . . . . . 9 |- (A C_ B -> ((x e. A /\ ph) -> ((x e. B /\ ph) /\ x e. A)))
10	9	eximdv 1622	. . . . . . . 8 |- (A C_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
11		eupick 2267	. . . . . . . . 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 423	. . . . . . . 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 66	. . . . . . 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 72	. . . . . 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 422	. . . . 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 461	. . . 4 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> ((x e. B /\ ph) -> x e. A))
17	16	exp3acom23 1372	. . 3 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> (ph -> (x e. B -> x e. A)))
18	17	imp 418	. 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 183	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 176   /\ wa 358  E.wex 1541   e. wcel 1710  E!weu 2204  E.wrex 2616  E!wreu 2617   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)