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Theorem euim 2193
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 2192 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
31, 2anim12ii 553 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  ( E. x ph  /\  E* x ph ) ) )
4 eu5 2181 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
53, 4syl6ibr 218 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 358   A.wal 1527   E.wex 1528   E!weu 2143   E*wmo 2144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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