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Theorem nf4dc 1633
Description: Variable  x is effectively not free in  ph iff  ph is always true or always false, given a decidability condition. The reverse direction, nf4r 1634, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4dc  |-  (DECID  E. x ph  ->  ( F/ x ph 
<->  ( A. x ph  \/  A. x  -.  ph ) ) )

Proof of Theorem nf4dc
StepHypRef Expression
1 nf2 1631 . . 3  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
2 imordc 867 . . 3  |-  (DECID  E. x ph  ->  ( ( E. x ph  ->  A. x ph )  <->  ( -.  E. x ph  \/  A. x ph ) ) )
31, 2syl5bb 191 . 2  |-  (DECID  E. x ph  ->  ( F/ x ph 
<->  ( -.  E. x ph  \/  A. x ph ) ) )
4 orcom 702 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
5 alnex 1460 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
65orbi2i 736 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
74, 6bitr4i 186 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
83, 7syl6bb 195 1  |-  (DECID  E. x ph  ->  ( F/ x ph 
<->  ( A. x ph  \/  A. x  -.  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 682  DECID wdc 804   A.wal 1314   F/wnf 1421   E.wex 1453
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-gen 1410  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-fal 1322  df-nf 1422
This theorem is referenced by: (None)
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