Theorem eupickb 2135

Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

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 2133	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
2	1	3adant2 1019	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3		3simpc 999	. . 3 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ph /\ ps ) ) )
4		pm3.22 265	. . . . 5 |- ( ( ph /\ ps ) -> ( ps /\ ph ) )
5	4	eximi 1623	. . . 4 |- ( E. x ( ph /\ ps ) -> E. x ( ps /\ ph ) )
6	5	anim2i 342	. . 3 |- ( ( E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ps /\ ph ) ) )
7		eupick 2133	. . 3 |- ( ( E! x ps /\ E. x ( ps /\ ph ) ) -> ( ps -> ph ) )
8	3, 6, 7	3syl 17	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )
9	2, 8	impbid 129	1 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   E.wex 1515   E!weu 2054

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by: (None)