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Theorem tbwlem2 1461
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
tbwlem2  |-  ( (
ph  ->  ( ps  ->  F.  ) )  ->  (
( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )

Proof of Theorem tbwlem2
StepHypRef Expression
1 tbw-ax4 1458 . . . . 5  |-  (  F. 
->  ch )
2 tbw-ax1 1455 . . . . . 6  |-  ( ( ps  ->  F.  )  ->  ( (  F.  ->  ch )  ->  ( ps  ->  ch ) ) )
3 tbwlem1 1460 . . . . . 6  |-  ( ( ( ps  ->  F.  )  ->  ( (  F. 
->  ch )  ->  ( ps  ->  ch ) ) )  ->  ( (  F.  ->  ch )  -> 
( ( ps  ->  F.  )  ->  ( ps  ->  ch ) ) ) )
42, 3ax-mp 8 . . . . 5  |-  ( (  F.  ->  ch )  ->  ( ( ps  ->  F.  )  ->  ( ps  ->  ch ) ) )
51, 4ax-mp 8 . . . 4  |-  ( ( ps  ->  F.  )  ->  ( ps  ->  ch ) )
6 tbwlem1 1460 . . . 4  |-  ( ( ( ps  ->  F.  )  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ( ps 
->  F.  )  ->  ch ) ) )
75, 6ax-mp 8 . . 3  |-  ( ps 
->  ( ( ps  ->  F.  )  ->  ch )
)
8 tbw-ax1 1455 . . 3  |-  ( (
ph  ->  ( ps  ->  F.  ) )  ->  (
( ( ps  ->  F.  )  ->  ch )  ->  ( ph  ->  ch ) ) )
9 tbw-ax1 1455 . . 3  |-  ( ( ps  ->  ( ( ps  ->  F.  )  ->  ch ) )  ->  (
( ( ( ps 
->  F.  )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
107, 8, 9mpsyl 59 . 2  |-  ( (
ph  ->  ( ps  ->  F.  ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
11 tbw-ax1 1455 . 2  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )
1210, 11tbwsyl 1459 1  |-  ( (
ph  ->  ( ps  ->  F.  ) )  ->  (
( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )
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
This theorem depends on definitions:  df-bi 177  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator