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Theorem merlem10 1573
Description: Step 19 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
merlem10 ((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓)))

Proof of Theorem merlem10
StepHypRef Expression
1 meredith 1563 . 2 (((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑)))
2 meredith 1563 . . 3 ((((((𝜑𝜓) → 𝜑) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → 𝜑) → ((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓))))
3 merlem9 1572 . . 3 (((((((𝜑𝜓) → 𝜑) → (¬ 𝜑 → ¬ 𝜃)) → 𝜑) → 𝜑) → ((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓)))) → ((((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑))) → ((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓)))))
42, 3ax-mp 5 . 2 ((((((𝜑𝜑) → (¬ 𝜑 → ¬ 𝜑)) → 𝜑) → 𝜑) → ((𝜑𝜑) → (𝜑𝜑))) → ((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓))))
51, 4ax-mp 5 1 ((𝜑 → (𝜑𝜓)) → (𝜃 → (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:  merlem11  1574
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