Theorem reupick 2269

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 2053	. . 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 558	. . . . . . . . . . 11 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
5		an23 484	. . . . . . . . . . 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 1285	. . . . . . . . 9 |- (A (_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
8		eupick 1427	. . . . . . . . . 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 1642	. . . . . . 7 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 1643	. . . . . . 7 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 481	. . . . . 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 514	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 955  E.wex 977  E!weu 1373  E.wrex 1638  E!wreu 1639   (_ wss 2037

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-reu 1643  df-in 2041  df-ss 2043

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