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Theorem euim 2331
Description: Add existential uniqueness 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 2330 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
31, 2anim12ii 554 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  ( E. x ph  /\  E* x ph ) ) )
4 eu5 2319 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
53, 4syl6ibr 219 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 359   A.wal 1549   E.wex 1550   E!weu 2281   E*wmo 2282
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 2285  df-mo 2286
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