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Theorem exmoeu 2280
Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exmoeu  |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )

Proof of Theorem exmoeu
StepHypRef Expression
1 df-mo 2243 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
21biimpi 187 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  E! x ph ) )
32com12 29 . 2  |-  ( E. x ph  ->  ( E* x ph  ->  E! x ph ) )
41biimpri 198 . . . 4  |-  ( ( E. x ph  ->  E! x ph )  ->  E* x ph )
5 euex 2261 . . . 4  |-  ( E! x ph  ->  E. x ph )
64, 5imim12i 55 . . 3  |-  ( ( E* x ph  ->  E! x ph )  -> 
( ( E. x ph  ->  E! x ph )  ->  E. x ph )
)
7 peirce 174 . . 3  |-  ( ( ( E. x ph  ->  E! x ph )  ->  E. x ph )  ->  E. x ph )
86, 7syl 16 . 2  |-  ( ( E* x ph  ->  E! x ph )  ->  E. x ph )
93, 8impbii 181 1  |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547   E!weu 2238   E*wmo 2239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243
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