Theorem exmoeudc 2117

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 2058	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
2	1	biimpi 120	. . 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 133	. . . 4 |- ( ( E. x ph -> E! x ph ) -> E* x ph )
5		euex 2084	. . . 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 916	. . 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 143	1 |- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 105  DECID wdc 836   E.wex 1515   E!weu 2054   E*wmo 2055

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-in2 616  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-dc 837  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by: (None)