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Theorem exmoeudc 2005
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 1946 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
21biimpi 118 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  E! x ph ) )
32com12 30 . 2  |-  ( E. x ph  ->  ( E* x ph  ->  E! x ph ) )
41biimpri 131 . . . 4  |-  ( ( E. x ph  ->  E! x ph )  ->  E* x ph )
5 euex 1972 . . . 4  |-  ( E! x ph  ->  E. x ph )
64, 5imim12i 58 . . 3  |-  ( ( E* x ph  ->  E! x ph )  -> 
( ( E. x ph  ->  E! x ph )  ->  E. x ph )
)
7 peircedc 854 . . 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 141 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 103  DECID wdc 776   E.wex 1422   E!weu 1942   E*wmo 1943
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 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by: (None)
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