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Theorem exmoeu 2322
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 2285 . . . 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 2303 . . . 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 1550   E!weu 2280   E*wmo 2281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285
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