Theorem reupick3 3489

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

Assertion

Ref	Expression
reupick3	|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )

Distinct variable group:   x, A

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

Proof of Theorem reupick3

Step	Hyp	Ref	Expression
1		df-reu 2515	. . . 4 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		df-rex 2514	. . . . 5 |- ( E. x e. A ( ph /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
3		anass 401	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
4	3	exbii 1651	. . . . 5 |- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
5	2, 4	bitr4i 187	. . . 4 |- ( E. x e. A ( ph /\ ps ) <-> E. x ( ( x e. A /\ ph ) /\ ps ) )
6		eupick 2157	. . . 4 |- ( ( E! x ( x e. A /\ ph ) /\ E. x ( ( x e. A /\ ph ) /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
7	1, 5, 6	syl2anb 291	. . 3 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
8	7	expd 258	. 2 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( x e. A -> ( ph -> ps ) ) )
9	8	3impia 1224	1 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   E.wex 1538   E!weu 2077   e. wcel 2200   E.wrex 2509   E!wreu 2510

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-rex 2514  df-reu 2515

This theorem is referenced by:  reupick2  3490