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Theorem luk-1 1577
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 meredith 1563 . 2 (((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
2 merlem13 1576 . . . 4 ((𝜑𝜓) → ((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓))
3 merlem13 1576 . . . 4 (((𝜑𝜓) → ((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓)) → ((((((𝜓𝜒) → (𝜑𝜒)) → 𝜑) → (¬ ¬ ¬ (𝜑𝜓) → ¬ (𝜑𝜓))) → ¬ ¬ (𝜑𝜓)) → ((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓)))
42, 3ax-mp 5 . . 3 ((((((𝜓𝜒) → (𝜑𝜒)) → 𝜑) → (¬ ¬ ¬ (𝜑𝜓) → ¬ (𝜑𝜓))) → ¬ ¬ (𝜑𝜓)) → ((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓))
5 meredith 1563 . . 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 5 . 2 ((((((𝜒𝜒) → (¬ ¬ ¬ 𝜑 → ¬ 𝜑)) → ¬ ¬ 𝜑) → 𝜓) → ((𝜓𝜒) → (𝜑𝜒))) → ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒))))
71, 6ax-mp 5 1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
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  1580  luklem2  1581  luklem4  1583  luklem6  1585  luklem7  1586  luklem8  1587
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