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Theorem tbwlem4 1463
Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbwlem4  |-  ( ( ( ph  ->  F.  )  ->  ps )  -> 
( ( ps  ->  F.  )  ->  ph ) )

Proof of Theorem tbwlem4
StepHypRef Expression
1 tbw-ax4 1458 . . . . 5  |-  (  F. 
->  F.  )
2 tbw-ax1 1455 . . . . . 6  |-  ( ( ps  ->  F.  )  ->  ( (  F.  ->  F.  )  ->  ( ps  ->  F.  ) ) )
3 tbwlem1 1460 . . . . . 6  |-  ( ( ( ps  ->  F.  )  ->  ( (  F. 
->  F.  )  ->  ( ps  ->  F.  ) )
)  ->  ( (  F.  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  ( ps  ->  F.  ) ) ) )
42, 3ax-mp 8 . . . . 5  |-  ( (  F.  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  ( ps  ->  F.  ) ) )
51, 4ax-mp 8 . . . 4  |-  ( ( ps  ->  F.  )  ->  ( ps  ->  F.  ) )
6 tbwlem1 1460 . . . 4  |-  ( ( ( ps  ->  F.  )  ->  ( ps  ->  F.  ) )  ->  ( ps  ->  ( ( ps 
->  F.  )  ->  F.  ) ) )
75, 6ax-mp 8 . . 3  |-  ( ps 
->  ( ( ps  ->  F.  )  ->  F.  )
)
8 tbw-ax1 1455 . . . 4  |-  ( ( ( ph  ->  F.  )  ->  ps )  -> 
( ( ps  ->  ( ( ps  ->  F.  )  ->  F.  ) )  ->  ( ( ph  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  F.  ) ) ) )
9 tbwlem1 1460 . . . 4  |-  ( ( ( ( ph  ->  F.  )  ->  ps )  ->  ( ( ps  ->  ( ( ps  ->  F.  )  ->  F.  ) )  ->  ( ( ph  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  F.  ) ) ) )  ->  ( ( ps 
->  ( ( ps  ->  F.  )  ->  F.  )
)  ->  ( (
( ph  ->  F.  )  ->  ps )  ->  (
( ph  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  F.  )
) ) ) )
108, 9ax-mp 8 . . 3  |-  ( ( ps  ->  ( ( ps  ->  F.  )  ->  F.  ) )  ->  (
( ( ph  ->  F.  )  ->  ps )  ->  ( ( ph  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  F.  ) ) ) )
117, 10ax-mp 8 . 2  |-  ( ( ( ph  ->  F.  )  ->  ps )  -> 
( ( ph  ->  F.  )  ->  ( ( ps  ->  F.  )  ->  F.  ) ) )
12 tbwlem2 1461 . . 3  |-  ( ( ( ph  ->  F.  )  ->  ( ( ps 
->  F.  )  ->  F.  ) )  ->  (
( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  (
( ps  ->  F.  )  ->  ph ) ) )
13 tbwlem3 1462 . . 3  |-  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  (
( ps  ->  F.  )  ->  ph ) )  -> 
( ( ps  ->  F.  )  ->  ph ) )
1412, 13tbwsyl 1459 . 2  |-  ( ( ( ph  ->  F.  )  ->  ( ( ps 
->  F.  )  ->  F.  ) )  ->  (
( ps  ->  F.  )  ->  ph ) )
1511, 14tbwsyl 1459 1  |-  ( ( ( ph  ->  F.  )  ->  ps )  -> 
( ( ps  ->  F.  )  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1308
This theorem is referenced by:  tbwlem5  1464  re1luk2  1466
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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