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Theorem luk-1 1420
 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 ((φψ) → ((ψχ) → (φχ)))

Proof of Theorem luk-1
StepHypRef Expression
1 ax-meredith 1406 . 2 (((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ) → ((ψχ) → (φχ)))
2 merlem13 1419 . . . 4 ((φψ) → ((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ))
3 merlem13 1419 . . . 4 (((φψ) → ((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ)) → ((((((ψχ) → (φχ)) → φ) → (¬ ¬ ¬ (φψ) → ¬ (φψ))) → ¬ ¬ (φψ)) → ((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ)))
42, 3ax-mp 8 . . 3 ((((((ψχ) → (φχ)) → φ) → (¬ ¬ ¬ (φψ) → ¬ (φψ))) → ¬ ¬ (φψ)) → ((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ))
5 ax-meredith 1406 . . 3 (((((((ψχ) → (φχ)) → φ) → (¬ ¬ ¬ (φψ) → ¬ (φψ))) → ¬ ¬ (φψ)) → ((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ)) → ((((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ) → ((ψχ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))
64, 5ax-mp 8 . 2 ((((((χχ) → (¬ ¬ ¬ φ → ¬ φ)) → ¬ ¬ φ) → ψ) → ((ψχ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))
71, 6ax-mp 8 1 ((φψ) → ((ψχ) → (φχ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 8  ax-meredith 1406 This theorem is referenced by:  luklem1  1423  luklem2  1424  luklem4  1426  luklem6  1428  luklem7  1429  luklem8  1430
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