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Theorem tbw-bijust 1458
Description: Justification for tbw-negdf 1459. (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 186 . 2  |-  ( (
ph 
<->  ps )  <->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
2 pm2.21 102 . . . . 5  |-  ( -.  ( ps  ->  ph )  ->  ( ( ps  ->  ph )  ->  F.  )
)
32imim2i 15 . . . 4  |-  ( ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  -> 
( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) ) )
4 id 21 . . . . . 6  |-  ( -.  ( ps  ->  ph )  ->  -.  ( ps  ->  ph ) )
5 falim 1325 . . . . . 6  |-  (  F. 
->  -.  ( ps  ->  ph ) )
64, 5ja 155 . . . . 5  |-  ( ( ( ps  ->  ph )  ->  F.  )  ->  -.  ( ps  ->  ph )
)
76imim2i 15 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ph )  ->  F.  ) )  ->  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
83, 7impbii 182 . . 3  |-  ( ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  <->  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) )
98notbii 289 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  <->  -.  (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
) )
10 pm2.21 102 . . 3  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  (
( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  ) )
11 ax-1 7 . . . . 5  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  (
( ( ( ph  ->  ps )  ->  (
( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
) ) )
12 falim 1325 . . . . 5  |-  (  F. 
->  ( ( ( (
ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )  ->  -.  ( ( ph  ->  ps )  ->  (
( ps  ->  ph )  ->  F.  ) ) ) )
1311, 12ja 155 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  ( ( ( ( ph  ->  ps )  ->  ( ( ps 
->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) ) )
1413pm2.43i 45 . . 3  |-  ( ( ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  ->  F.  )  ->  -.  ( ( ph  ->  ps )  -> 
( ( ps  ->  ph )  ->  F.  )
) )
1510, 14impbii 182 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  ) )  <->  ( (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )
)
161, 9, 153bitri 264 1  |-  ( (
ph 
<->  ps )  <->  ( (
( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  F.  )
)  ->  F.  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    F. wfal 1313
This theorem is referenced by:  tbw-negdf  1459
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-tru 1315  df-fal 1316
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