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Theorem merlem13 1576
Description: Step 35 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
merlem13 ((𝜑𝜓) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓))

Proof of Theorem merlem13
StepHypRef Expression
1 merlem12 1575 . . . . 5 (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))
2 merlem12 1575 . . . . . . . 8 (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)
3 merlem5 1568 . . . . . . . 8 ((((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑))
42, 3ax-mp 5 . . . . . . 7 (¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)
5 merlem6 1569 . . . . . . 7 ((¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑) → ((((¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))))
64, 5ax-mp 5 . . . . . 6 ((((¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)))
7 meredith 1563 . . . . . 6 (((((¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓) → (¬ ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → ((((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))))
86, 7ax-mp 5 . . . . 5 ((((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)))
91, 8ax-mp 5 . . . 4 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))
10 merlem6 1569 . . . 4 ((¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑)) → ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)))
119, 10ax-mp 5 . . 3 ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑))
12 merlem11 1574 . . 3 (((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)) → ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑))
1311, 12ax-mp 5 . 2 ((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑)
14 meredith 1563 . 2 (((((𝜓𝜓) → (¬ 𝜑 → ¬ ((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑))) → 𝜑) → 𝜑) → ((𝜑𝜓) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓)))
1513, 14ax-mp 5 1 ((𝜑𝜓) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓))
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:  luk-1  1577
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