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Theorem merlem11 1725
 Description: Step 20 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
merlem11 ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem merlem11
StepHypRef Expression
1 meredith 1714 . 2 (((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑)))
2 merlem10 1724 . . 3 ((𝜑 → (𝜑𝜓)) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜓)))
3 merlem10 1724 . . 3 (((𝜑 → (𝜑𝜓)) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))) → ((((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑))) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))))
42, 3ax-mp 5 . 2 ((((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑))) → ((𝜑 → (𝜑𝜓)) → (𝜑𝜓)))
51, 4ax-mp 5 1 ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 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:  merlem12  1726  merlem13  1727  luk-2  1729  luk-3  1730
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