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Theorem alexim 1632
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1606. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim  |-  ( A. x ph  ->  -.  E. x  -.  ph )

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 611 . . . . 5  |-  ( ph  ->  ( -.  ph  -> F.  ) )
21alimi 1442 . . . 4  |-  ( A. x ph  ->  A. x
( -.  ph  -> F.  ) )
3 exim 1586 . . . 4  |-  ( A. x ( -.  ph  -> F.  )  ->  ( E. x  -.  ph  ->  E. x F.  ) )
42, 3syl 14 . . 3  |-  ( A. x ph  ->  ( E. x  -.  ph  ->  E. x F.  ) )
5 nfv 1515 . . . 4  |-  F/ x F.
6519.9 1631 . . 3  |-  ( E. x F.  <-> F.  )
74, 6syl6ib 160 . 2  |-  ( A. x ph  ->  ( E. x  -.  ph  -> F.  )
)
8 dfnot 1360 . 2  |-  ( -. 
E. x  -.  ph  <->  ( E. x  -.  ph  -> F.  ) )
97, 8sylibr 133 1  |-  ( A. x ph  ->  -.  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1340   F. wfal 1347   E.wex 1479
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-in1 604  ax-in2 605  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448
This theorem is referenced by:  exnalim  1633  exists2  2110
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