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Theorem merco1lem3 1473
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1468. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem3  |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  ( ch  ->  ph ) )

Proof of Theorem merco1lem3
StepHypRef Expression
1 merco1lem2 1472 . . 3  |-  ( ( ( ph  ->  ph )  ->  F.  )  ->  (
( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  F.  ) )
2 retbwax2 1471 . . . 4  |-  ( ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  -> 
( ph  ->  ph )
)  ->  ( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) ) )
3 merco1lem2 1472 . . . 4  |-  ( ( ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
)  ->  ( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) ) )  -> 
( ( ( (
ph  ->  ph )  ->  F.  )  ->  ( ( (
ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  F.  ) )  -> 
( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph ) ) ) ) )
42, 3ax-mp 8 . . 3  |-  ( ( ( ( ph  ->  ph )  ->  F.  )  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  F.  ) )  -> 
( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph ) ) ) )
51, 4ax-mp 8 . 2  |-  ( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) )
6 merco1lem2 1472 . . 3  |-  ( ( ( ch  ->  ph )  ->  F.  )  ->  (
( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  F.  ) )
7 retbwax2 1471 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  -> 
( ch  ->  ph )
)  ->  ( ( ph  ->  ( ( (
ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) )  ->  (
( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  -> 
( ch  ->  ph )
) ) )
8 merco1lem2 1472 . . . 4  |-  ( ( ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  ( ch  ->  ph )
)  ->  ( ( ph  ->  ( ( (
ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) )  ->  (
( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  -> 
( ch  ->  ph )
) ) )  -> 
( ( ( ( ch  ->  ph )  ->  F.  )  ->  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  F.  ) )  ->  (
( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph ) ) )  ->  ( ( (
ph  ->  ps )  -> 
( ch  ->  F.  ) )  ->  ( ch  ->  ph ) ) ) ) )
97, 8ax-mp 8 . . 3  |-  ( ( ( ( ch  ->  ph )  ->  F.  )  ->  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  F.  ) )  -> 
( ( ph  ->  ( ( ( ph  ->  ph )  ->  ( ph  ->  F.  ) )  -> 
( ph  ->  ph )
) )  ->  (
( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  -> 
( ch  ->  ph )
) ) )
106, 9ax-mp 8 . 2  |-  ( (
ph  ->  ( ( (
ph  ->  ph )  ->  ( ph  ->  F.  ) )  ->  ( ph  ->  ph )
) )  ->  (
( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  -> 
( ch  ->  ph )
) )
115, 10ax-mp 8 1  |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  F.  ) )  ->  ( ch  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    F. wfal 1308
This theorem is referenced by:  merco1lem4  1474  merco1lem6  1476  merco1lem11  1482  merco1lem12  1483  merco1lem18  1489
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
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