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Theorem tbwlem3 1462
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
tbwlem3  |-  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps )

Proof of Theorem tbwlem3
StepHypRef Expression
1 tbw-ax3 1457 . . 3  |-  ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )
2 tbw-ax2 1456 . . . 4  |-  ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( (
ph  ->  F.  )  ->  ph )  ->  ph ) ) )
3 tbw-ax1 1455 . . . 4  |-  ( ( ( ( ( (
ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )
)  ->  ( (
( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) ) )
42, 3tbwsyl 1459 . . 3  |-  ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) ) )
51, 4ax-mp 8 . 2  |-  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )
6 tbw-ax1 1455 . . 3  |-  ( ( ( ( ( (
ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )  ->  ( ( ( ( ( ( (
ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps )  ->  ps )  ->  (
( ( ( (
ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps )
) )
7 tbw-ax3 1457 . . 3  |-  ( ( ( ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )  -> 
( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )
86, 7tbwsyl 1459 . 2  |-  ( ( ( ( ( (
ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )  ->  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps ) )
95, 8ax-mp 8 1  |-  ( ( ( ( ( ph  ->  F.  )  ->  ph )  ->  ph )  ->  ps )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1308
This theorem is referenced by:  tbwlem4  1463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
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