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Theorem tbwlem5 1483
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
tbwlem5  |-  ( ( ( ph  ->  ( ps  ->  F.  ) )  ->  F.  )  ->  ph )

Proof of Theorem tbwlem5
StepHypRef Expression
1 tbw-ax2 1475 . . . 4  |-  ( ph  ->  ( ps  ->  ph )
)
2 tbw-ax1 1474 . . . 4  |-  ( ( ps  ->  ph )  -> 
( ( ph  ->  F.  )  ->  ( ps  ->  F.  ) ) )
31, 2tbwsyl 1478 . . 3  |-  ( ph  ->  ( ( ph  ->  F.  )  ->  ( ps  ->  F.  ) ) )
4 tbwlem1 1479 . . 3  |-  ( (
ph  ->  ( ( ph  ->  F.  )  ->  ( ps  ->  F.  ) )
)  ->  ( ( ph  ->  F.  )  ->  (
ph  ->  ( ps  ->  F.  ) ) ) )
53, 4ax-mp 8 . 2  |-  ( (
ph  ->  F.  )  ->  (
ph  ->  ( ps  ->  F.  ) ) )
6 tbwlem4 1482 . 2  |-  ( ( ( ph  ->  F.  )  ->  ( ph  ->  ( ps  ->  F.  )
) )  ->  (
( ( ph  ->  ( ps  ->  F.  )
)  ->  F.  )  ->  ph ) )
75, 6ax-mp 8 1  |-  ( ( ( ph  ->  ( ps  ->  F.  ) )  ->  F.  )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1326
This theorem is referenced by:  re1luk3  1486
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 178  df-tru 1328  df-fal 1329
  Copyright terms: Public domain W3C validator