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Theorem merco1lem2 1491
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1487. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem2  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  ch ) )

Proof of Theorem merco1lem2
StepHypRef Expression
1 retbwax2 1490 . . 3  |-  ( ( ( ( ps  ->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  )  ->  ( ( ch  ->  ph )  -> 
( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  ) ) )
2 merco1 1487 . . 3  |-  ( ( ( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  )  ->  (
( ch  ->  ph )  ->  ( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  ) ) )  ->  ( ( ( ( ch  ->  ph )  ->  ( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  ) )  ->  ps )  ->  ( ph  ->  ps ) ) )
31, 2ax-mp 8 . 2  |-  ( ( ( ( ch  ->  ph )  ->  ( (
( ps  ->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  ) )  ->  ps )  ->  ( ph  ->  ps ) )
4 merco1 1487 . 2  |-  ( ( ( ( ( ch 
->  ph )  ->  (
( ( ps  ->  ta )  ->  ( ph  ->  F.  ) )  ->  F.  ) )  ->  ps )  ->  ( ph  ->  ps ) )  ->  (
( ( ph  ->  ps )  ->  ch )  ->  ( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  ch ) ) )
53, 4ax-mp 8 1  |-  ( ( ( ph  ->  ps )  ->  ch )  -> 
( ( ( ps 
->  ta )  ->  ( ph  ->  F.  ) )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1326
This theorem is referenced by:  merco1lem3  1492  merco1lem10  1500  merco1lem11  1501  merco1lem18  1508
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
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