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Theorem tbwlem1 1460
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
tbwlem1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )

Proof of Theorem tbwlem1
StepHypRef Expression
1 tbw-ax2 1456 . . . 4  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ps )
)
2 tbw-ax1 1455 . . . 4  |-  ( ( ( ps  ->  ch )  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
31, 2tbwsyl 1459 . . 3  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
4 tbw-ax1 1455 . . . 4  |-  ( ( ( ps  ->  ch )  ->  ( ( ps 
->  ch )  ->  ch ) )  ->  (
( ( ( ps 
->  ch )  ->  ch )  ->  ch )  -> 
( ( ps  ->  ch )  ->  ch )
) )
5 tbw-ax3 1457 . . . 4  |-  ( ( ( ( ( ps 
->  ch )  ->  ch )  ->  ch )  -> 
( ( ps  ->  ch )  ->  ch )
)  ->  ( ( ps  ->  ch )  ->  ch ) )
64, 5tbwsyl 1459 . . 3  |-  ( ( ( ps  ->  ch )  ->  ( ( ps 
->  ch )  ->  ch ) )  ->  (
( ps  ->  ch )  ->  ch ) )
73, 6tbwsyl 1459 . 2  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ch )
)
8 tbw-ax1 1455 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ( ps  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) ) )
9 tbw-ax1 1455 . 2  |-  ( ( ps  ->  ( ( ps  ->  ch )  ->  ch ) )  ->  (
( ( ( ps 
->  ch )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
107, 8, 9mpsyl 59 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  tbwlem2  1461  tbwlem4  1463  tbwlem5  1464  re1luk3  1467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
  Copyright terms: Public domain W3C validator