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Theorem merlem12 1408
Description: Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem12  |-  ( ( ( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )

Proof of Theorem merlem12
StepHypRef Expression
1 merlem5 1401 . . . 4  |-  ( ( ch  ->  ch )  ->  ( -.  -.  ch  ->  ch ) )
2 merlem2 1398 . . . 4  |-  ( ( ( ch  ->  ch )  ->  ( -.  -.  ch  ->  ch ) )  ->  ( th  ->  ( -.  -.  ch  ->  ch ) ) )
31, 2ax-mp 8 . . 3  |-  ( th 
->  ( -.  -.  ch  ->  ch ) )
4 merlem4 1400 . . 3  |-  ( ( th  ->  ( -.  -.  ch  ->  ch )
)  ->  ( (
( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ( ( ( th 
->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )
) )
53, 4ax-mp 8 . 2  |-  ( ( ( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ( ( ( th 
->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )
)
6 merlem11 1407 . 2  |-  ( ( ( ( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ( ( ( th 
->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )
)  ->  ( (
( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph ) )
75, 6ax-mp 8 1  |-  ( ( ( th  ->  ( -.  -.  ch  ->  ch ) )  ->  ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem13  1409
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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