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Theorem merlem10 1576
Description: Step 19 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
merlem10 |- ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) )
Proof of Theorem merlem10
StepHypRef Expression
1 meredith 1566 . 2 |- ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) )
2 meredith 1566 . . 3 |- ( ( ( ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> -. th ) ) -> ph ) -> ph ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) )
3 merlem9 1575 . . 3 |- ( ( ( ( ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> -. th ) ) -> ph ) -> ph ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) ) -> ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) ) )
42, 3ax-mp 5 . 2 |- ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) )
51, 4ax-mp 5 1 |- ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by:  merlem11  1577
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