Theorem eupickb 2538

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

Assertion

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

Proof of Theorem eupickb

Step	Hyp	Ref	Expression
1		eupick 2536	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
2	1	3adant2 1080	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3		exancom 1787	. . . 4 |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4		eupick 2536	. . . 4 |- ( ( E! x ps /\ E. x ( ps /\ ph ) ) -> ( ps -> ph ) )
5	3, 4	sylan2b 492	. . 3 |- ( ( E! x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )
6	5	3adant1 1079	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )
7	2, 6	impbid 202	1 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)