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Theorem euim 2068
Description: Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
euim  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )

Proof of Theorem euim
StepHypRef Expression
1 ax-1 6 . . 3  |-  ( E. x ph  ->  ( E! x ps  ->  E. x ph ) )
2 euimmo 2067 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
31, 2anim12ii 341 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  ( E. x ph  /\  E* x ph ) ) )
4 eu5 2047 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
53, 4syl6ibr 161 1  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1330   E.wex 1469   E!weu 2000   E*wmo 2001
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004
This theorem is referenced by: (None)
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