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Theorem luklem2 1414
Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
luklem2  |-  ( (
ph  ->  -.  ps )  ->  ( ( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )

Proof of Theorem luklem2
StepHypRef Expression
1 luk-1 1410 . . 3  |-  ( (
ph  ->  -.  ps )  ->  ( ( -.  ps  ->  ch )  ->  ( ph  ->  ch ) ) )
2 luk-3 1412 . . . 4  |-  ( ps 
->  ( -.  ps  ->  ch ) )
3 luk-1 1410 . . . 4  |-  ( ( ps  ->  ( -.  ps  ->  ch ) )  ->  ( ( ( -.  ps  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
42, 3ax-mp 8 . . 3  |-  ( ( ( -.  ps  ->  ch )  ->  ( ph  ->  ch ) )  -> 
( ps  ->  ( ph  ->  ch ) ) )
51, 4luklem1 1413 . 2  |-  ( (
ph  ->  -.  ps )  ->  ( ps  ->  ( ph  ->  ch ) ) )
6 luk-1 1410 . 2  |-  ( ( ps  ->  ( ph  ->  ch ) )  -> 
( ( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )
75, 6luklem1 1413 1  |-  ( (
ph  ->  -.  ps )  ->  ( ( ( ph  ->  ch )  ->  th )  ->  ( ps  ->  th )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  luklem3  1415  luklem6  1418  ax3  1423
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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