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	⊢ (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ)))))

Proof of Theorem merlem9

Step	Hyp	Ref	Expression
1		merlem6 1412	. . . 4 ⊢ ((θ → (ψ → τ)) → (((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)))
2		merlem8 1414	. . . 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 5	. . 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		ax-meredith 1406	. . 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 5	. 2 ⊢ (((((χ → (θ → (ψ → τ))) → ¬ η) → (¬ ψ → ¬ η)) → ψ) → (φ → ψ))
6		ax-meredith 1406	. 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 5	1 ⊢ (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ)))))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-meredith 1406

This theorem is referenced by:  merlem10  1416