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Theorem alexim 1577
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1551. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim  |-  ( A. x ph  ->  -.  E. x  -.  ph )

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 584 . . . . 5  |-  ( ph  ->  ( -.  ph  -> F.  ) )
21alimi 1385 . . . 4  |-  ( A. x ph  ->  A. x
( -.  ph  -> F.  ) )
3 exim 1531 . . . 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 1462 . . . 4  |-  F/ x F.
6519.9 1576 . . 3  |-  ( E. x F.  <-> F.  )
74, 6syl6ib 159 . 2  |-  ( A. x ph  ->  ( E. x  -.  ph  -> F.  )
)
8 dfnot 1303 . 2  |-  ( -. 
E. x  -.  ph  <->  ( E. x  -.  ph  -> F.  ) )
97, 8sylibr 132 1  |-  ( A. x ph  ->  -.  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1283   F. wfal 1290   E.wex 1422
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-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by:  exnalim  1578  exists2  2039
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