Theorem reupick 2874

Description: Restricted uniqueness "picks" a member of a subclass.

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

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 2615	. . 3 |- (A C_ B -> (x e. A -> x e. B))
2	1	ad2antrr 440	. 2 |- (((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A -> x e. B))
3	1	ancrd 323	. . . . . . . . . . . 12 |- (A C_ B -> (x e. A -> (x e. B /\ x e. A)))
4	3	anim1d 619	. . . . . . . . . . 11 |- (A C_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
5		an23 543	. . . . . . . . . . 11 |- (((x e. B /\ x e. A) /\ ph) <-> ((x e. B /\ ph) /\ x e. A))
6	4, 5	syl6ib 229	. . . . . . . . . 10 |- (A C_ B -> ((x e. A /\ ph) -> ((x e. B /\ ph) /\ x e. A)))
7	6	eximdv 1669	. . . . . . . . 9 |- (A C_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
8		eupick 1834	. . . . . . . . . 10 |- ((E!x(x e. B /\ ph) /\ E.x((x e. B /\ ph) /\ x e. A)) -> ((x e. B /\ ph) -> x e. A))
9	8	ex 402	. . . . . . . . 9 |- (E!x(x e. B /\ ph) -> (E.x((x e. B /\ ph) /\ x e. A) -> ((x e. B /\ ph) -> x e. A)))
10	7, 9	syl9 71	. . . . . . . 8 |- (A C_ B -> (E!x(x e. B /\ ph) -> (E.x(x e. A /\ ph) -> ((x e. B /\ ph) -> x e. A))))
11	10	com23 36	. . . . . . 7 |- (A C_ B -> (E.x(x e. A /\ ph) -> (E!x(x e. B /\ ph) -> ((x e. B /\ ph) -> x e. A))))
12	11	imp32 390	. . . . . 6 |- ((A C_ B /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph))) -> ((x e. B /\ ph) -> x e. A))
13		df-rex 2110	. . . . . . 7 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 2111	. . . . . . 7 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 540	. . . . . 6 |- ((E.x e. A ph /\ E!x e. B ph) <-> (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph)))
16	12, 15	sylan2b 501	. . . . 5 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> ((x e. B /\ ph) -> x e. A))
17	16	exp3a 405	. . . 4 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> (x e. B -> (ph -> x e. A)))
18	17	com23 36	. . 3 |- ((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> (ph -> (x e. B -> x e. A)))
19	18	imp 377	. 2 |- (((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. B -> x e. A))
20	2, 19	impbid 574	1 |- (((A C_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A <-> x e. B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  E.wex 1326  E!weu 1771  E.wrex 2106  E!wreu 2107   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-reu 2111  df-in 2603  df-ss 2605

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