Theorem merlem9 653

Description: Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

Ref	Expression
merlem9	⊢ (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ)))))

Proof of Theorem merlem9

Step	Hyp	Ref	Expression
1		merlem6 650	. . . 4 ⊢ ((θ → (ψ → τ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)))
2		merlem8 652	. . . 4 ⊢ (((θ → (ψ → τ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η))) → ((((ψ → τ) → (¬ (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η))))
3	1, 2	ax-mp 6	. . 3 ⊢ ((((ψ → τ) → (¬ (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)))
4		meredith 644	. . 3 ⊢ (((((ψ → τ) → (¬ (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ) → ¬ φ)) → (¬ (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ¬ θ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η))) → (((((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φ → ψ)))
5	3, 4	ax-mp 6	. 2 ⊢ (((((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φ → ψ))
6		meredith 644	. 2 ⊢ ((((((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φ → ψ)) → (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ))))))
7	5, 6	ax-mp 6	1 ⊢ (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ)))))

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  merlem10 654

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

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