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Theorem alexdc 1555
Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1581. (Contributed by Jim Kingdon, 2-Jun-2018.)
Assertion
Ref Expression
alexdc  |-  ( A. xDECID  ph 
->  ( A. x ph  <->  -. 
E. x  -.  ph ) )

Proof of Theorem alexdc
StepHypRef Expression
1 nfa1 1479 . . 3  |-  F/ x A. xDECID 
ph
2 notnotbdc 804 . . . 4  |-  (DECID  ph  ->  (
ph 
<->  -.  -.  ph )
)
32sps 1475 . . 3  |-  ( A. xDECID  ph 
->  ( ph  <->  -.  -.  ph ) )
41, 3albid 1551 . 2  |-  ( A. xDECID  ph 
->  ( A. x ph  <->  A. x  -.  -.  ph ) )
5 alnex 1433 . 2  |-  ( A. x  -.  -.  ph  <->  -.  E. x  -.  ph )
64, 5syl6bb 194 1  |-  ( A. xDECID  ph 
->  ( A. x ph  <->  -. 
E. x  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 780   A.wal 1287   E.wex 1426
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395
This theorem is referenced by: (None)
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