Theorem reupick2 3493

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reupick2	|- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reupick2

Step	Hyp	Ref	Expression
1		ancr 321	. . . . . 6 |- ( ( ps -> ph ) -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi 2595	. . . . 5 |- ( A. x e. A ( ps -> ph ) -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2626	. . . . 5 |- ( A. x e. A ( ps -> ( ph /\ ps ) ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
4	2, 3	syl 14	. . . 4 |- ( A. x e. A ( ps -> ph ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
5		reupick3 3492	. . . . . 6 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )
6	5	3exp 1228	. . . . 5 |- ( E! x e. A ph -> ( E. x e. A ( ph /\ ps ) -> ( x e. A -> ( ph -> ps ) ) ) )
7	6	com12 30	. . . 4 |- ( E. x e. A ( ph /\ ps ) -> ( E! x e. A ph -> ( x e. A -> ( ph -> ps ) ) ) )
8	4, 7	syl6 33	. . 3 |- ( A. x e. A ( ps -> ph ) -> ( E. x e. A ps -> ( E! x e. A ph -> ( x e. A -> ( ph -> ps ) ) ) ) )
9	8	3imp1 1246	. 2 |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph -> ps ) )
10		rsp 2579	. . . 4 |- ( A. x e. A ( ps -> ph ) -> ( x e. A -> ( ps -> ph ) ) )
11	10	3ad2ant1 1044	. . 3 |- ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) -> ( x e. A -> ( ps -> ph ) ) )
12	11	imp 124	. 2 |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ps -> ph ) )
13	9, 12	impbid 129	1 |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   e. wcel 2202   A.wral 2510   E.wrex 2511   E!wreu 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-ral 2515  df-rex 2516  df-reu 2517

This theorem is referenced by: (None)