Theorem eupickb

Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994) (Proof shortened by Wolf Lammen, 27-Dec-2018)

	Ref	Expression
Assertion	eupickb	|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		eupick	|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
2	1	3adant2	|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3		exancom	|- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4		eupick	|- ( ( E! x ps /\ E. x ( ps /\ ph ) ) -> ( ps -> ph ) )
5	3 4	sylan2b	|- ( ( E! x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )
6	5	3adant1	|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )
7	2 6	impbid	|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )