Theorem reupick 2275

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

Assertion

Ref	Expression
reupick	|- (((A (_ 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 2059	. . 3 |- (A (_ B -> (x e. A -> x e. B))
2	1	ad2antrr 404	. 2 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A -> x e. B))
3	1	ancrd 299	. . . . . . . . . . . 12 |- (A (_ B -> (x e. A -> (x e. B /\ x e. A)))
4	3	anim1d 559	. . . . . . . . . . 11 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
5		an23 485	. . . . . . . . . . 11 |- (((x e. B /\ x e. A) /\ ph) <-> ((x e. B /\ ph) /\ x e. A))
6	4, 5	syl6ib 212	. . . . . . . . . 10 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ ph) /\ x e. A)))
7	6	19.22dv 1288	. . . . . . . . 9 |- (A (_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
8		eupick 1432	. . . . . . . . . 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 373	. . . . . . . . 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 57	. . . . . . . 8 |- (A (_ B -> (E!x(x e. B /\ ph) -> (E.x(x e. A /\ ph) -> ((x e. B /\ ph) -> x e. A))))
11	10	com23 32	. . . . . . 7 |- (A (_ B -> (E.x(x e. A /\ ph) -> (E!x(x e. B /\ ph) -> ((x e. B /\ ph) -> x e. A))))
12	11	imp32 363	. . . . . 6 |- ((A (_ B /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph))) -> ((x e. B /\ ph) -> x e. A))
13		df-rex 1647	. . . . . . 7 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 1648	. . . . . . 7 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 482	. . . . . 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 452	. . . . 5 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> ((x e. B /\ ph) -> x e. A))
17	16	exp3a 375	. . . 4 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> (x e. B -> (ph -> x e. A)))
18	17	com23 32	. . 3 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> (ph -> (x e. B -> x e. A)))
19	18	imp 350	. 2 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. B -> x e. A))
20	2, 19	impbid 515	1 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A <-> x e. B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956  E.wex 978  E!weu 1378  E.wrex 1643  E!wreu 1644   (_ wss 2043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-rex 1647  df-reu 1648  df-in 2047  df-ss 2049

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