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Theorem exmoeu 2311
Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
Assertion
Ref Expression
exmoeu  |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )

Proof of Theorem exmoeu
StepHypRef Expression
1 df-mo 2280 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
21biimpi 194 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  E! x ph ) )
32com12 31 . 2  |-  ( E. x ph  ->  ( E* x ph  ->  E! x ph ) )
4 exmo 2304 . . . . 5  |-  ( E. x ph  \/  E* x ph )
54ori 375 . . . 4  |-  ( -. 
E. x ph  ->  E* x ph )
65con1i 129 . . 3  |-  ( -. 
E* x ph  ->  E. x ph )
7 euex 2303 . . 3  |-  ( E! x ph  ->  E. x ph )
86, 7ja 161 . 2  |-  ( ( E* x ph  ->  E! x ph )  ->  E. x ph )
93, 8impbii 188 1  |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   E.wex 1596   E!weu 2275   E*wmo 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719
This theorem depends on definitions:  df-bi 185  df-or 370  df-ex 1597  df-eu 2279  df-mo 2280
This theorem is referenced by: (None)
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