Theorem reupick 1578

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 group(s):   x,A   x,B

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 1502	. . 3 |- (A (_ B -> (x e. A -> x e. B))
2	1	ad2antll 320	. 2 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A -> x e. B))
3	1	ancrd 247	. . . . . . . . . . . 12 |- (A (_ B -> (x e. A -> (x e. B /\ x e. A)))
4	3	anim1d 432	. . . . . . . . . . 11 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
5		an23 371	. . . . . . . . . . 11 |- (((x e. B /\ x e. A) /\ ph) <-> ((x e. B /\ ph) /\ x e. A))
6	4, 5	syl6ib 185	. . . . . . . . . 10 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ ph) /\ x e. A)))
7	6	19.22dv 947	. . . . . . . . 9 |- (A (_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
8		eupick 1055	. . . . . . . . . 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	exp 291	. . . . . . . . 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 55	. . . . . . . 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 281	. . . . . 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 1206	. . . . . . 7 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 1207	. . . . . . 7 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 369	. . . . . 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 347	. . . . 5 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> ((x e. B /\ ph) -> x e. A))
17	16	exp3a 292	. . . 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 277	. 2 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. B -> x e. A))
20	2, 19	impbid 397	1 |- (((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) /\ ph) -> (x e. A <-> x e. B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678  E!weu 1007   e. wcel 1092  E.wrex 1202  E!wreu 1203   (_ wss 1487

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-in 1491  df-ss 1492

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