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Theorem luk-1 1410
Description: 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
luk-1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )

Proof of Theorem luk-1
StepHypRef Expression
1 ax-meredith 1396 . 2  |-  ( ( ( ( ( ch 
->  ch )  ->  ( -.  -.  -.  ph  ->  -. 
ph ) )  ->  -.  -.  ph )  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
2 merlem13 1409 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( ( ( ch  ->  ch )  ->  ( -.  -.  -.  ph 
->  -.  ph ) )  ->  -.  -.  ph )  ->  ps ) )
3 merlem13 1409 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ( ( ( ch  ->  ch )  ->  ( -.  -.  -.  ph  ->  -.  ph )
)  ->  -.  -.  ph )  ->  ps ) )  ->  ( ( ( ( ( ( ps 
->  ch )  ->  ( ph  ->  ch ) )  ->  ph )  ->  ( -.  -.  -.  ( ph  ->  ps )  ->  -.  ( ph  ->  ps )
) )  ->  -.  -.  ( ph  ->  ps ) )  ->  (
( ( ( ch 
->  ch )  ->  ( -.  -.  -.  ph  ->  -. 
ph ) )  ->  -.  -.  ph )  ->  ps ) ) )
42, 3ax-mp 8 . . 3  |-  ( ( ( ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  ->  ph )  ->  ( -.  -.  -.  ( ph  ->  ps )  ->  -.  ( ph  ->  ps ) ) )  ->  -.  -.  ( ph  ->  ps ) )  ->  (
( ( ( ch 
->  ch )  ->  ( -.  -.  -.  ph  ->  -. 
ph ) )  ->  -.  -.  ph )  ->  ps ) )
5 ax-meredith 1396 . . 3  |-  ( ( ( ( ( ( ( ps  ->  ch )  ->  ( ph  ->  ch ) )  ->  ph )  ->  ( -.  -.  -.  ( ph  ->  ps )  ->  -.  ( ph  ->  ps ) ) )  ->  -.  -.  ( ph  ->  ps ) )  ->  (
( ( ( ch 
->  ch )  ->  ( -.  -.  -.  ph  ->  -. 
ph ) )  ->  -.  -.  ph )  ->  ps ) )  ->  (
( ( ( ( ( ch  ->  ch )  ->  ( -.  -.  -.  ph  ->  -.  ph )
)  ->  -.  -.  ph )  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )  ->  ( ( ph  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) ) ) )
64, 5ax-mp 8 . 2  |-  ( ( ( ( ( ( ch  ->  ch )  ->  ( -.  -.  -.  ph 
->  -.  ph ) )  ->  -.  -.  ph )  ->  ps )  ->  (
( ps  ->  ch )  ->  ( ph  ->  ch ) ) )  -> 
( ( ph  ->  ps )  ->  ( ( ps  ->  ch )  -> 
( ph  ->  ch )
) ) )
71, 6ax-mp 8 1  |-  ( (
ph  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  luklem1  1413  luklem2  1414  luklem4  1416  luklem6  1418  luklem7  1419  luklem8  1420
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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