Theorem exmoeudc 2011

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 1952	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
2	1	biimpi 118	. . 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 131	. . . 4 |- ( ( E. x ph -> E! x ph ) -> E* x ph )
5		euex 1978	. . . 4 |- ( E! x ph -> E. x ph )
6	4, 5	imim12i 58	. . 3 |- ( ( E* x ph -> E! x ph ) -> ( ( E. x ph -> E! x ph ) -> E. x ph ) )
7		peircedc 858	. . 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 141	1 |- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 103  DECID wdc 780   E.wex 1426   E!weu 1948   E*wmo 1949

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-in2 580  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-dc 781  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952

This theorem is referenced by: (None)