Theorem exmoeudc 2063

Description: Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)

Assertion

Ref	Expression
exmoeudc	|- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )

Proof of Theorem exmoeudc

Step	Hyp	Ref	Expression
1		df-mo 2004	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
2	1	biimpi 119	. . 3 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
3	2	com12 30	. 2 |- ( E. x ph -> ( E* x ph -> E! x ph ) )
4	1	biimpri 132	. . . 4 |- ( ( E. x ph -> E! x ph ) -> E* x ph )
5		euex 2030	. . . 4 |- ( E! x ph -> E. x ph )
6	4, 5	imim12i 59	. . 3 |- ( ( E* x ph -> E! x ph ) -> ( ( E. x ph -> E! x ph ) -> E. x ph ) )
7		peircedc 900	. . 3 |- (DECID E. x ph -> ( ( ( E. x ph -> E! x ph ) -> E. x ph ) -> E. x ph ) )
8	6, 7	syl5 32	. 2 |- (DECID E. x ph -> ( ( E* x ph -> E! x ph ) -> E. x ph ) )
9	3, 8	impbid2 142	1 |- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 104  DECID wdc 820   E.wex 1469   E!weu 2000   E*wmo 2001

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-in2 605  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-dc 821  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by: (None)