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Theorem merlem11 1407
Description: Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem11  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )

Proof of Theorem merlem11
StepHypRef Expression
1 ax-meredith 1396 . 2  |-  ( ( ( ( ( ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )
2 merlem10 1406 . . 3  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  (
( ph  ->  ( ph  ->  ps ) )  -> 
( ph  ->  ps )
) )
3 merlem10 1406 . . 3  |-  ( ( ( ph  ->  ( ph  ->  ps ) )  ->  ( ( ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) )  ->  ( (
( ( ( (
ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )  ->  ( ( ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) ) )
42, 3ax-mp 8 . 2  |-  ( ( ( ( ( (
ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )  ->  ( ( ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) )
51, 4ax-mp 8 1  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem12  1408  merlem13  1409  luk-2  1411  luk-3  1412
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
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