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Theorem merlem3 1399
Description: Step 7 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
merlem3  |-  ( ( ( ps  ->  ch )  ->  ph )  ->  ( ch  ->  ph ) )

Proof of Theorem merlem3
StepHypRef Expression
1 merlem2 1398 . . . 4  |-  ( ( ( -.  ch  ->  -. 
ch )  ->  ( -.  ch  ->  -.  ch )
)  ->  ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
) )
2 merlem2 1398 . . . 4  |-  ( ( ( ( -.  ch  ->  -.  ch )  -> 
( -.  ch  ->  -. 
ch ) )  -> 
( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
) )  ->  (
( ( ( ch 
->  ph )  ->  ( -.  ps  ->  -.  ps )
)  ->  ps )  ->  ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
) ) )
31, 2ax-mp 8 . . 3  |-  ( ( ( ( ch  ->  ph )  ->  ( -.  ps  ->  -.  ps )
)  ->  ps )  ->  ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
) )
4 ax-meredith 1396 . . 3  |-  ( ( ( ( ( ch 
->  ph )  ->  ( -.  ps  ->  -.  ps )
)  ->  ps )  ->  ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
) )  ->  (
( ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
)  ->  ch )  ->  ( ps  ->  ch ) ) )
53, 4ax-mp 8 . 2  |-  ( ( ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
)  ->  ch )  ->  ( ps  ->  ch ) )
6 ax-meredith 1396 . 2  |-  ( ( ( ( ( ph  ->  ph )  ->  ( -.  ch  ->  -.  ch )
)  ->  ch )  ->  ( ps  ->  ch ) )  ->  (
( ( ps  ->  ch )  ->  ph )  -> 
( ch  ->  ph )
) )
75, 6ax-mp 8 1  |-  ( ( ( ps  ->  ch )  ->  ph )  ->  ( ch  ->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem4  1400  merlem6  1402
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator