Theorem merlem10 654

Description: Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

Ref	Expression
merlem10	⊢ ((φ → (φ → ψ)) → (θ → (φ → ψ)))

Proof of Theorem merlem10

Step	Hyp	Ref	Expression
1		meredith 644	. 2 ⊢ (((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ)))
2		meredith 644	. . 3 ⊢ ((((((φ → ψ) → φ) → (¬ φ → ¬ θ)) → φ) → φ) → ((φ → (φ → ψ)) → (θ → (φ → ψ))))
3		merlem9 653	. . 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 6	. 2 ⊢ ((((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ))) → ((φ → (φ → ψ)) → (θ → (φ → ψ))))
5	1, 4	ax-mp 6	1 ⊢ ((φ → (φ → ψ)) → (θ → (φ → ψ)))

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  merlem11 655

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

metamath.org