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Theorem euim 2016
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 5 . . 3  |-  ( E. x ph  ->  ( E! x ps  ->  E. x ph ) )
2 euimmo 2015 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
31, 2anim12ii 335 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  ( E. x ph  /\  E* x ph ) ) )
4 eu5 1995 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
53, 4syl6ibr 160 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 102   A.wal 1287   E.wex 1426   E!weu 1948   E*wmo 1949
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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