Theorem reupick2 3449

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 2560	. . . . 5 |- ( A. x e. A ( ps -> ph ) -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim 2591	. . . . 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 3448	. . . . . 6 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )
6	5	3exp 1204	. . . . 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 1222	. 2 |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph -> ps ) )
10		rsp 2544	. . . 4 |- ( A. x e. A ( ps -> ph ) -> ( x e. A -> ( ps -> ph ) ) )
11	10	3ad2ant1 1020	. . 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 980   e. wcel 2167   A.wral 2475   E.wrex 2476   E!wreu 2477

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-ral 2480  df-rex 2481  df-reu 2482

This theorem is referenced by: (None)