Theorem merlem10 1416

Description: Step 19 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
merlem10	⊢ ((φ → (φ → ψ)) → (θ → (φ → ψ)))

Proof of Theorem merlem10

Step	Hyp	Ref	Expression
1		ax-meredith 1406	. 2 ⊢ (((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ)))
2		ax-meredith 1406	. . 3 ⊢ ((((((φ → ψ) → φ) → (¬ φ → ¬ θ)) → φ) → φ) → ((φ → (φ → ψ)) → (θ → (φ → ψ))))
3		merlem9 1415	. . 3 ⊢ (((((((φ → ψ) → φ) → (¬ φ → ¬ θ)) → φ) → φ) → ((φ → (φ → ψ)) → (θ → (φ → ψ)))) → ((((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ))) → ((φ → (φ → ψ)) → (θ → (φ → ψ)))))
4	2, 3	ax-mp 5	. 2 ⊢ ((((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ))) → ((φ → (φ → ψ)) → (θ → (φ → ψ))))
5	1, 4	ax-mp 5	1 ⊢ ((φ → (φ → ψ)) → (θ → (φ → ψ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-meredith 1406

This theorem is used by:  merlem11  1417