MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tbw-negdf Unicode version

Theorem tbw-negdf 1454
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 pm2.21 100 . . 3  |-  ( -. 
ph  ->  ( ph  ->  F.  ) )
2 ax-1 5 . . . . 5  |-  ( -. 
ph  ->  ( ( ph  ->  F.  )  ->  -.  ph ) )
3 falim 1319 . . . . 5  |-  (  F. 
->  ( ( ph  ->  F.  )  ->  -.  ph )
)
42, 3ja 153 . . . 4  |-  ( (
ph  ->  F.  )  ->  ( ( ph  ->  F.  )  ->  -.  ph ) )
54pm2.43i 43 . . 3  |-  ( (
ph  ->  F.  )  ->  -. 
ph )
61, 5impbii 180 . 2  |-  ( -. 
ph 
<->  ( ph  ->  F.  ) )
7 tbw-bijust 1453 . 2  |-  ( ( -.  ph  <->  ( ph  ->  F.  ) )  <->  ( (
( -.  ph  ->  (
ph  ->  F.  ) )  ->  ( ( ( ph  ->  F.  )  ->  -.  ph )  ->  F.  )
)  ->  F.  )
)
86, 7mpbi 199 1  |-  ( ( ( -.  ph  ->  (
ph  ->  F.  ) )  ->  ( ( ( ph  ->  F.  )  ->  -.  ph )  ->  F.  )
)  ->  F.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    F. wfal 1308
This theorem is referenced by:  re1luk2  1466  re1luk3  1467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator