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Theorem luk-2 1578
Description: 2 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-2 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem luk-2
StepHypRef Expression
1 merlem5 1568 . . . . 5 ((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑)))
2 merlem4 1567 . . . . 5 (((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑)))
31, 2ax-mp 5 . . . 4 ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑))
4 merlem11 1574 . . . 4 (((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑)) → ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑))
53, 4ax-mp 5 . . 3 ((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑)
6 meredith 1563 . . 3 (((((𝜑 → ¬ (¬ 𝜑𝜑)) → (¬ ¬ 𝜑 → ¬ (¬ 𝜑𝜑))) → ¬ 𝜑) → ¬ 𝜑) → ((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)))
75, 6ax-mp 5 . 2 ((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑))
8 merlem11 1574 . 2 (((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)) → ((¬ 𝜑𝜑) → 𝜑))
97, 8ax-mp 5 1 ((¬ 𝜑𝜑) → 𝜑)
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:  luklem4  1583
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