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Theorem tbw-negdf 1576
Description: The definition of negation, in terms of  -> and F.. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbw-negdf  |-  ( ( ( -.  ph  ->  (
ph  -> F.  ) )  ->  ( ( (
ph  -> F.  )  ->  -.  ph )  -> F.  ) )  -> F.  )

Proof of Theorem tbw-negdf
StepHypRef Expression
1 pm2.21 111 . . 3  |-  ( -. 
ph  ->  ( ph  -> F.  ) )
2 ax-1 6 . . . . 5  |-  ( -. 
ph  ->  ( ( ph  -> F.  )  ->  -.  ph ) )
3 falim 1451 . . . . 5  |-  ( F. 
->  ( ( ph  -> F.  )  ->  -.  ph )
)
42, 3ja 164 . . . 4  |-  ( (
ph  -> F.  )  -> 
( ( ph  -> F.  )  ->  -.  ph )
)
54pm2.43i 49 . . 3  |-  ( (
ph  -> F.  )  ->  -.  ph )
61, 5impbii 190 . 2  |-  ( -. 
ph 
<->  ( ph  -> F.  ) )
7 tbw-bijust 1575 . 2  |-  ( ( -.  ph  <->  ( ph  -> F.  ) )  <->  ( (
( -.  ph  ->  (
ph  -> F.  ) )  ->  ( ( (
ph  -> F.  )  ->  -.  ph )  -> F.  ) )  -> F.  ) )
86, 7mpbi 211 1  |-  ( ( ( -.  ph  ->  (
ph  -> F.  ) )  ->  ( ( (
ph  -> F.  )  ->  -.  ph )  -> F.  ) )  -> F.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187   F. wfal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-tru 1440  df-fal 1443
This theorem is referenced by:  re1luk2  1588  re1luk3  1589
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