Theorem reupick2 3541

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 532	. . . . . 6 |- ((ps -> ph) -> (ps -> (ph /\ ps)))
2	1	ralimi 2689	. . . . 5 |- (A.x e. A (ps -> ph) -> A.x e. A (ps -> (ph /\ ps)))
3		rexim 2718	. . . . 5 |- (A.x e. A (ps -> (ph /\ ps)) -> (E.x e. A ps -> E.x e. A (ph /\ ps)))
4	2, 3	syl 15	. . . 4 |- (A.x e. A (ps -> ph) -> (E.x e. A ps -> E.x e. A (ph /\ ps)))
5		reupick3 3540	. . . . . 6 |- ((E!x e. A ph /\ E.x e. A (ph /\ ps) /\ x e. A) -> (ph -> ps))
6	5	3exp 1150	. . . . 5 |- (E!x e. A ph -> (E.x e. A (ph /\ ps) -> (x e. A -> (ph -> ps))))
7	6	com12 27	. . . 4 |- (E.x e. A (ph /\ ps) -> (E!x e. A ph -> (x e. A -> (ph -> ps))))
8	4, 7	syl6 29	. . 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 1164	. 2 |- (((A.x e. A (ps -> ph) /\ E.x e. A ps /\ E!x e. A ph) /\ x e. A) -> (ph -> ps))
10		rsp 2674	. . . 4 |- (A.x e. A (ps -> ph) -> (x e. A -> (ps -> ph)))
11	10	3ad2ant1 976	. . 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 418	. 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 183	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 176   /\ wa 358   /\ w3a 934   e. wcel 1710  A.wral 2614  E.wrex 2615  E!wreu 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rex 2620  df-reu 2621

This theorem is referenced by: (None)