Theorem merlem12 656

Description: Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.

Assertion

Ref	Expression
merlem12	⊢ (((θ → (¬ ¬ χ → χ)) → φ) → φ)

Proof of Theorem merlem12

Step	Hyp	Ref	Expression
1		merlem5 649	. . . 4 ⊢ ((χ → χ) → (¬ ¬ χ → χ))
2		merlem2 646	. . . 4 ⊢ (((χ → χ) → (¬ ¬ χ → χ)) → (θ → (¬ ¬ χ → χ)))
3	1, 2	ax-mp 6	. . 3 ⊢ (θ → (¬ ¬ χ → χ))
4		merlem4 648	. . 3 ⊢ ((θ → (¬ ¬ χ → χ)) → (((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ)))
5	3, 4	ax-mp 6	. 2 ⊢ (((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ))
6		merlem11 655	. 2 ⊢ ((((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ)) → (((θ → (¬ ¬ χ → χ)) → φ) → φ))
7	5, 6	ax-mp 6	1 ⊢ (((θ → (¬ ¬ χ → χ)) → φ) → φ)

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  merlem13 657

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

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