Theorem exmoeudc 1963

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 1904	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
2	1	biimpi 113	. . 3 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
3	2	com12 27	. 2 |- ( E. x ph -> ( E* x ph -> E! x ph ) )
4	1	biimpri 124	. . . 4 |- ( ( E. x ph -> E! x ph ) -> E* x ph )
5		euex 1930	. . . 4 |- ( E! x ph -> E. x ph )
6	4, 5	imim12i 53	. . 3 |- ( ( E* x ph -> E! x ph ) -> ( ( E. x ph -> E! x ph ) -> E. x ph ) )
7		peircedc 820	. . 3 |- (DECID E. x ph -> ( ( ( E. x ph -> E! x ph ) -> E. x ph ) -> E. x ph ) )
8	6, 7	syl5 28	. 2 |- (DECID E. x ph -> ( ( E* x ph -> E! x ph ) -> E. x ph ) )
9	3, 8	impbid2 131	1 |- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 98  DECID wdc 742   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-dc 743  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by: (None)