Theorem merlem11 655

Description: Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

Ref	Expression
merlem11	⊢ ((φ → (φ → ψ)) → (φ → ψ))

Proof of Theorem merlem11

Step	Hyp	Ref	Expression
1		meredith 644	. 2 ⊢ (((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ)))
2		merlem10 654	. . 3 ⊢ ((φ → (φ → ψ)) → ((φ → (φ → ψ)) → (φ → ψ)))
3		merlem10 654	. . 3 ⊢ (((φ → (φ → ψ)) → ((φ → (φ → ψ)) → (φ → ψ))) → ((((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ))) → ((φ → (φ → ψ)) → (φ → ψ))))
4	2, 3	ax-mp 6	. 2 ⊢ ((((((φ → φ) → (¬ φ → ¬ φ)) → φ) → φ) → ((φ → φ) → (φ → φ))) → ((φ → (φ → ψ)) → (φ → ψ)))
5	1, 4	ax-mp 6	1 ⊢ ((φ → (φ → ψ)) → (φ → ψ))

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  merlem12 656  merlem13 657  luk-2 659  luk-3 660

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

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