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Theorem exmoeudc 2069
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
StepHypRef Expression
1 df-mo 2010 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
21biimpi 119 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  E! x ph ) )
32com12 30 . 2  |-  ( E. x ph  ->  ( E* x ph  ->  E! x ph ) )
41biimpri 132 . . . 4  |-  ( ( E. x ph  ->  E! x ph )  ->  E* x ph )
5 euex 2036 . . . 4  |-  ( E! x ph  ->  E. x ph )
64, 5imim12i 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 ) )
86, 7syl5 32 . 2  |-  (DECID  E. x ph  ->  ( ( E* x ph  ->  E! x ph )  ->  E. x ph ) )
93, 8impbid2 142 1  |-  (DECID  E. x ph  ->  ( E. x ph 
<->  ( E* x ph  ->  E! x ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104  DECID wdc 820   E.wex 1472   E!weu 2006   E*wmo 2007
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010
This theorem is referenced by: (None)
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