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Theorem merlem3 1566
Description: Step 7 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
merlem3 (((𝜓𝜒) → 𝜑) → (𝜒𝜑))

Proof of Theorem merlem3
StepHypRef Expression
1 merlem2 1565 . . . 4 (((¬ 𝜒 → ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜒)) → ((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒)))
2 merlem2 1565 . . . 4 ((((¬ 𝜒 → ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜒)) → ((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒))) → ((((𝜒𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒))))
31, 2ax-mp 5 . . 3 ((((𝜒𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒)))
4 meredith 1563 . . 3 (((((𝜒𝜑) → (¬ 𝜓 → ¬ 𝜓)) → 𝜓) → ((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒))) → ((((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓𝜒)))
53, 4ax-mp 5 . 2 ((((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓𝜒))
6 meredith 1563 . 2 (((((𝜑𝜑) → (¬ 𝜒 → ¬ 𝜒)) → 𝜒) → (𝜓𝜒)) → (((𝜓𝜒) → 𝜑) → (𝜒𝜑)))
75, 6ax-mp 5 1 (((𝜓𝜒) → 𝜑) → (𝜒𝜑))
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:  merlem4  1567  merlem6  1569
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