ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euim Unicode version

Theorem euim 2013
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 2012 . . 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 1992 . 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 1285   E.wex 1424   E!weu 1945   E*wmo 1946
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator