Theorem eupickb 2081

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 2079	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
2	1	3adant2 1001	. 2 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
3		3simpc 981	. . 3 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ph /\ ps ) ) )
4		pm3.22 263	. . . . 5 |- ( ( ph /\ ps ) -> ( ps /\ ph ) )
5	4	eximi 1580	. . . 4 |- ( E. x ( ph /\ ps ) -> E. x ( ps /\ ph ) )
6	5	anim2i 340	. . 3 |- ( ( E! x ps /\ E. x ( ph /\ ps ) ) -> ( E! x ps /\ E. x ( ps /\ ph ) ) )
7		eupick 2079	. . 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 128	1 |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   E.wex 1469   E!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by: (None)