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Theorem nf4dc 1576
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 1577, 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 1574 . . 3  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
2 imordc 807 . . 3  |-  (DECID  E. x ph  ->  ( ( E. x ph  ->  A. x ph )  <->  ( -.  E. x ph  \/  A. x ph ) ) )
31, 2syl5bb 185 . 2  |-  (DECID  E. x ph  ->  ( F/ x ph 
<->  ( -.  E. x ph  \/  A. x ph ) ) )
4 orcom 657 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
5 alnex 1404 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
65orbi2i 689 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
74, 6bitr4i 180 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
83, 7syl6bb 189 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 102    \/ wo 639  DECID wdc 753   A.wal 1257   F/wnf 1365   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265  df-nf 1366
This theorem is referenced by: (None)
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