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Theorem merlem7 1403
Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem7  |-  ( ph  ->  ( ( ( ps 
->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
) )

Proof of Theorem merlem7
StepHypRef Expression
1 merlem4 1400 . 2  |-  ( ( ps  ->  ch )  ->  ( ( ( ps 
->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
) )
2 merlem6 1402 . . . 4  |-  ( ( ( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th )  ->  (
( ( ( ( ps  ->  ch )  ->  th )  ->  (
( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th ) )  ->  -.  ph )  ->  ( -.  ch  ->  -.  ph )
) )
3 ax-meredith 1396 . . . 4  |-  ( ( ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )  ->  ( ( ( ( ( ps  ->  ch )  ->  th )  ->  (
( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th ) )  ->  -.  ph )  ->  ( -.  ch  ->  -.  ph )
) )  ->  (
( ( ( ( ( ( ps  ->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
)  ->  -.  ph )  ->  ( -.  ch  ->  -. 
ph ) )  ->  ch )  ->  ( ps 
->  ch ) ) )
42, 3ax-mp 8 . . 3  |-  ( ( ( ( ( ( ( ps  ->  ch )  ->  th )  ->  (
( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th ) )  ->  -.  ph )  ->  ( -.  ch  ->  -.  ph )
)  ->  ch )  ->  ( ps  ->  ch ) )
5 ax-meredith 1396 . . 3  |-  ( ( ( ( ( ( ( ( ps  ->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
)  ->  -.  ph )  ->  ( -.  ch  ->  -. 
ph ) )  ->  ch )  ->  ( ps 
->  ch ) )  -> 
( ( ( ps 
->  ch )  ->  (
( ( ps  ->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
) )  ->  ( ph  ->  ( ( ( ps  ->  ch )  ->  th )  ->  (
( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th ) ) ) ) )
64, 5ax-mp 8 . 2  |-  ( ( ( ps  ->  ch )  ->  ( ( ( ps  ->  ch )  ->  th )  ->  (
( ( ch  ->  ta )  ->  ( -.  th 
->  -.  ps ) )  ->  th ) ) )  ->  ( ph  ->  ( ( ( ps  ->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
) ) )
71, 6ax-mp 8 1  |-  ( ph  ->  ( ( ( ps 
->  ch )  ->  th )  ->  ( ( ( ch 
->  ta )  ->  ( -.  th  ->  -.  ps )
)  ->  th )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem8  1404
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator