Theorem reupick 2290

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 2074	. . 3 |- (A (_ B -> (x e. A -> x e. B))
2	1	ad2antrr 406	. 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 563	. . . . . . . . . . 11 |- (A (_ B -> ((x e. A /\ ph) -> ((x e. B /\ x e. A) /\ ph)))
5		an23 488	. . . . . . . . . . 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 1294	. . . . . . . . 9 |- (A (_ B -> (E.x(x e. A /\ ph) -> E.x((x e. B /\ ph) /\ x e. A)))
8		eupick 1438	. . . . . . . . . 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 1657	. . . . . . 7 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 1658	. . . . . . 7 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 485	. . . . . 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 455	. . . . 5 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> ((x e. B /\ ph) -> x e. A))
17	16	exp3a 376	. . . 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 519	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 962  E.wex 984  E!weu 1384  E.wrex 1653  E!wreu 1654   (_ wss 2058

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-rex 1657  df-reu 1658  df-in 2062  df-ss 2064

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