Theorem reupick3

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

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

Proof

Step	Hyp	Ref	Expression
1		df-reu	|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		df-rex	|- ( E. x e. A ( ph /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
3		anass	|- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
4	3	exbii	|- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
5	2 4	bitr4i	|- ( E. x e. A ( ph /\ ps ) <-> E. x ( ( x e. A /\ ph ) /\ ps ) )
6		eupick	|- ( ( E! x ( x e. A /\ ph ) /\ E. x ( ( x e. A /\ ph ) /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
7	1 5 6	syl2anb	|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
8	7	expd	|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( x e. A -> ( ph -> ps ) ) )
9	8	3impia	|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )