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Theorem merlem13 1419
 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 1418 . . . . 5 (((θ → (¬ ¬ χχ)) → ¬ ((θ → (¬ ¬ χχ)) → ¬ ¬ φ)) → ¬ ((θ → (¬ ¬ χχ)) → ¬ ¬ φ))
2 merlem12 1418 . . . . . . . 8 (((θ → (¬ ¬ χχ)) → ¬ ¬ φ) → ¬ ¬ φ)
3 merlem5 1411 . . . . . . . 8 ((((θ → (¬ ¬ χχ)) → ¬ ¬ φ) → ¬ ¬ φ) → (¬ ¬ ((θ → (¬ ¬ χχ)) → ¬ ¬ φ) → ¬ ¬ φ))
42, 3ax-mp 5 . . . . . . 7 (¬ ¬ ((θ → (¬ ¬ χχ)) → ¬ ¬ φ) → ¬ ¬ φ)
5 merlem6 1412 . . . . . . 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 ax-meredith 1406 . . . . . 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 1412 . . . 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 1417 . . 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 ax-meredith 1406 . 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-meredith 1406 This theorem is referenced by:  luk-1  1420
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