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Theorem tbw-bijust 1453
Description: Justification for tbw-negdf 1454. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbw-bijust  |-  ( (
ph 
<->  ps )  <->  ( (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )
)

Proof of Theorem tbw-bijust
StepHypRef Expression
1 dfbi1 184 . 2  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
2 pm2.21 100 . . . . 5  |-  ( -.  ( ps  ->  ph )  ->  ( ( ps  ->  ph )  ->  F.  )
)
32imim2i 13 . . . 4  |-  ( ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  -> 
( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) ) )
4 id 19 . . . . . 6  |-  ( -.  ( ps  ->  ph )  ->  -.  ( ps  ->  ph ) )
5 falim 1319 . . . . . 6  |-  (  F. 
->  -.  ( ps  ->  ph ) )
64, 5ja 153 . . . . 5  |-  ( ( ( ps  ->  ph )  ->  F.  )  ->  -.  ( ps  ->  ph )
)
76imim2i 13 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ph )  ->  F.  ) )  ->  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
83, 7impbii 180 . . 3  |-  ( ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  <->  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) )
98notbii 287 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  <->  -.  (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
) )
10 pm2.21 100 . . 3  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  (
( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  ) )
11 ax-1 5 . . . . 5  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  (
( ( ( ph  ->  ps )  ->  (
( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
) ) )
12 falim 1319 . . . . 5  |-  (  F. 
->  ( ( ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )  ->  -.  ( ( ph  ->  ps )  ->  (
( ps  ->  ph )  ->  F.  ) ) ) )
1311, 12ja 153 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  ( ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) ) )
1413pm2.43i 43 . . 3  |-  ( ( ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) )
1510, 14impbii 180 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  <->  ( (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )
)
161, 9, 153bitri 262 1  |-  ( (
ph 
<->  ps )  <->  ( (
( ph  ->  ps )  ->  ( ( ps  ->  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:  tbw-negdf  1454
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
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