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Theorem luk-1 1580
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 meredith 1566 . 2 |- ( ( ( ( ( ch -> ch ) -> ( -. -. -. ph -> -. ph ) ) -> -. -. ph ) -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
2 merlem13 1579 . . . 4 |- ( ( ph -> ps ) -> ( ( ( ( ch -> ch ) -> ( -. -. -. ph -> -. ph ) ) -> -. -. ph ) -> ps ) )
3 merlem13 1579 . . . 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 5 . . 3 |- ( ( ( ( ( ( ps -> ch ) -> ( ph -> ch ) ) -> ph ) -> ( -. -. -. ( ph -> ps ) -> -. ( ph -> ps ) ) ) -> -. -. ( ph -> ps ) ) -> ( ( ( ( ch -> ch ) -> ( -. -. -. ph -> -. ph ) ) -> -. -. ph ) -> ps ) )
5 meredith 1566 . . 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 5 . 2 |- ( ( ( ( ( ( ch -> ch ) -> ( -. -. -. ph -> -. ph ) ) -> -. -. ph ) -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) )
71, 6ax-mp 5 1 |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
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:  luklem1  1583  luklem2  1584  luklem4  1586  luklem6  1588  luklem7  1589  luklem8  1590
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