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Theorem merlem1 1397
Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem1  |-  ( ( ( ch  ->  ( -.  ph  ->  ps )
)  ->  ta )  ->  ( ph  ->  ta ) )

Proof of Theorem merlem1
StepHypRef Expression
1 ax-meredith 1396 . . 3  |-  ( ( ( ( ( -. 
ph  ->  ps )  -> 
( -.  ( -. 
ta  ->  -.  ch )  ->  -.  -.  ( -. 
ph  ->  ps ) ) )  ->  ( -.  ta  ->  -.  ch )
)  ->  ta )  ->  ( ( ta  ->  -. 
ph )  ->  ( -.  ( -.  ph  ->  ps )  ->  -.  ph )
) )
2 ax-meredith 1396 . . 3  |-  ( ( ( ( ( ( -.  ph  ->  ps )  ->  ( -.  ( -. 
ta  ->  -.  ch )  ->  -.  -.  ( -. 
ph  ->  ps ) ) )  ->  ( -.  ta  ->  -.  ch )
)  ->  ta )  ->  ( ( ta  ->  -. 
ph )  ->  ( -.  ( -.  ph  ->  ps )  ->  -.  ph )
) )  ->  (
( ( ( ta 
->  -.  ph )  -> 
( -.  ( -. 
ph  ->  ps )  ->  -.  ph ) )  -> 
( -.  ph  ->  ps ) )  ->  ( ch  ->  ( -.  ph  ->  ps ) ) ) )
31, 2ax-mp 8 . 2  |-  ( ( ( ( ta  ->  -. 
ph )  ->  ( -.  ( -.  ph  ->  ps )  ->  -.  ph )
)  ->  ( -.  ph 
->  ps ) )  -> 
( ch  ->  ( -.  ph  ->  ps )
) )
4 ax-meredith 1396 . 2  |-  ( ( ( ( ( ta 
->  -.  ph )  -> 
( -.  ( -. 
ph  ->  ps )  ->  -.  ph ) )  -> 
( -.  ph  ->  ps ) )  ->  ( ch  ->  ( -.  ph  ->  ps ) ) )  ->  ( ( ( ch  ->  ( -.  ph 
->  ps ) )  ->  ta )  ->  ( ph  ->  ta ) ) )
53, 4ax-mp 8 1  |-  ( ( ( ch  ->  ( -.  ph  ->  ps )
)  ->  ta )  ->  ( ph  ->  ta ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem2  1398  merlem5  1401  luk-3  1412
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator