Theorem reupick2 3913

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 572	. . . . . 6 |- ( ( ps -> ph ) -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi 2952	. . . . 5 |- ( A. x e. A ( ps -> ph ) -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 3008	. . . . 5 |- ( A. x e. A ( ps -> ( ph /\ ps ) ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
4	2, 3	syl 17	. . . 4 |- ( A. x e. A ( ps -> ph ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
5		reupick3 3912	. . . . . 6 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )
6	5	3exp 1264	. . . . 5 |- ( E! x e. A ph -> ( E. x e. A ( ph /\ ps ) -> ( x e. A -> ( ph -> ps ) ) ) )
7	6	com12 32	. . . 4 |- ( E. x e. A ( ph /\ ps ) -> ( E! x e. A ph -> ( x e. A -> ( ph -> ps ) ) ) )
8	4, 7	syl6 35	. . 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 1280	. 2 |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph -> ps ) )
10		rsp 2929	. . . 4 |- ( A. x e. A ( ps -> ph ) -> ( x e. A -> ( ps -> ph ) ) )
11	10	3ad2ant1 1082	. . 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 445	. 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 202	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   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rex 2918  df-reu 2919

This theorem is referenced by:  grpoidval  27367  grpoidinv2  27369  grpoinv  27379