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Theorem luklem4 1416
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
luklem4  |-  ( ( ( ( -.  ph  ->  ph )  ->  ph )  ->  ps )  ->  ps )

Proof of Theorem luklem4
StepHypRef Expression
1 luk-2 1411 . . . 4  |-  ( ( -.  ( ( -. 
ph  ->  ph )  ->  ph )  ->  ( ( -.  ph  ->  ph )  ->  ph )
)  ->  ( ( -.  ph  ->  ph )  ->  ph ) )
2 luk-2 1411 . . . . 5  |-  ( ( -.  ph  ->  ph )  ->  ph )
3 luklem3 1415 . . . . 5  |-  ( ( ( -.  ph  ->  ph )  ->  ph )  -> 
( ( ( -.  ( ( -.  ph  ->  ph )  ->  ph )  ->  ( ( -.  ph  ->  ph )  ->  ph )
)  ->  ( ( -.  ph  ->  ph )  ->  ph ) )  ->  ( -.  ps  ->  ( ( -.  ph  ->  ph )  ->  ph ) ) ) )
42, 3ax-mp 8 . . . 4  |-  ( ( ( -.  ( ( -.  ph  ->  ph )  ->  ph )  ->  (
( -.  ph  ->  ph )  ->  ph ) )  ->  ( ( -. 
ph  ->  ph )  ->  ph )
)  ->  ( -.  ps  ->  ( ( -. 
ph  ->  ph )  ->  ph )
) )
51, 4ax-mp 8 . . 3  |-  ( -. 
ps  ->  ( ( -. 
ph  ->  ph )  ->  ph )
)
6 luk-1 1410 . . 3  |-  ( ( -.  ps  ->  (
( -.  ph  ->  ph )  ->  ph ) )  ->  ( ( ( ( -.  ph  ->  ph )  ->  ph )  ->  ps )  ->  ( -. 
ps  ->  ps ) ) )
75, 6ax-mp 8 . 2  |-  ( ( ( ( -.  ph  ->  ph )  ->  ph )  ->  ps )  ->  ( -.  ps  ->  ps )
)
8 luk-2 1411 . 2  |-  ( ( -.  ps  ->  ps )  ->  ps )
97, 8luklem1 1413 1  |-  ( ( ( ( -.  ph  ->  ph )  ->  ph )  ->  ps )  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  luklem5  1417  luklem6  1418  ax3  1423
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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