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Theorem merlem9 1415
 Description: Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem9 (((φψ) → (χ → (θ → (ψτ)))) → (η → (χ → (θ → (ψτ)))))

Proof of Theorem merlem9
StepHypRef Expression
1 merlem6 1412 . . . 4 ((θ → (ψτ)) → (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)))
2 merlem8 1414 . . . 4 (((θ → (ψτ)) → (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η))) → ((((ψτ) → (¬ (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η))))
31, 2ax-mp 8 . . 3 ((((ψτ) → (¬ (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)))
4 ax-meredith 1406 . . 3 (((((ψτ) → (¬ (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η))) → (((((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φψ)))
53, 4ax-mp 8 . 2 (((((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φψ))
6 ax-meredith 1406 . 2 ((((((χ → (θ → (ψτ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φψ)) → (((φψ) → (χ → (θ → (ψτ)))) → (η → (χ → (θ → (ψτ))))))
75, 6ax-mp 8 1 (((φψ) → (χ → (θ → (ψτ)))) → (η → (χ → (θ → (ψτ)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 8  ax-meredith 1406 This theorem is referenced by:  merlem10  1416
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