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Theorem nf4r 1602
Description: If  ph is always true or always false, then variable  x is effectively not free in 
ph. The converse holds given a decidability condition, as seen at nf4dc 1601. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4r  |-  ( ( A. x ph  \/  A. x  -.  ph )  ->  F/ x ph )

Proof of Theorem nf4r
StepHypRef Expression
1 orcom 680 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
2 alnex 1429 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
32orbi2i 712 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
41, 3bitr4i 185 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
5 imorr 831 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  ->  ( E. x ph  ->  A. x ph )
)
6 nf2 1599 . . 3  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
75, 6sylibr 132 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  ->  F/ x ph )
84, 7sylbir 133 1  |-  ( ( A. x ph  \/  A. x  -.  ph )  ->  F/ x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662   A.wal 1283   F/wnf 1390   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-io 663  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-4 1441  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: (None)
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