Theorem eupickb 2029

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 2027	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
2	1	3adant2 962	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3		3simpc 942	. . 3 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ph /\ ps ) ) )
4		pm3.22 261	. . . . 5 |- ( ( ph /\ ps ) -> ( ps /\ ph ) )
5	4	eximi 1536	. . . 4 |- ( E. x ( ph /\ ps ) -> E. x ( ps /\ ph ) )
6	5	anim2i 334	. . 3 |- ( ( E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ps /\ ph ) ) )
7		eupick 2027	. . 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 127	1 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   E.wex 1426   E!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952

This theorem is referenced by: (None)