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Theorem nf4 1812
Description: Variable  x is effectively not free in  ph iff  ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf4  |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )

Proof of Theorem nf4
StepHypRef Expression
1 nf2 1810 . 2  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
2 imor 401 . 2  |-  ( ( E. x ph  ->  A. x ph )  <->  ( -.  E. x ph  \/  A. x ph ) )
3 orcom 376 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
4 alnex 1533 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
54orbi2i 505 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
63, 5bitr4i 243 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
71, 2, 63bitri 262 1  |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357   A.wal 1530   E.wex 1531   F/wnf 1534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1532  df-nf 1535
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