Theorem merlem12 1418

Description: Step 28 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
merlem12	⊢ (((θ → (¬ ¬ χ → χ)) → φ) → φ)

Proof of Theorem merlem12

Step	Hyp	Ref	Expression
1		merlem5 1411	. . . 4 ⊢ ((χ → χ) → (¬ ¬ χ → χ))
2		merlem2 1408	. . . 4 ⊢ (((χ → χ) → (¬ ¬ χ → χ)) → (θ → (¬ ¬ χ → χ)))
3	1, 2	ax-mp 5	. . 3 ⊢ (θ → (¬ ¬ χ → χ))
4		merlem4 1410	. . 3 ⊢ ((θ → (¬ ¬ χ → χ)) → (((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ)))
5	3, 4	ax-mp 5	. 2 ⊢ (((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ))
6		merlem11 1417	. 2 ⊢ ((((θ → (¬ ¬ χ → χ)) → φ) → (((θ → (¬ ¬ χ → χ)) → φ) → φ)) → (((θ → (¬ ¬ χ → χ)) → φ) → φ))
7	5, 6	ax-mp 5	1 ⊢ (((θ → (¬ ¬ χ → χ)) → φ) → φ)

Syntax hints:  ¬ wn 3   → wi 4

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

This theorem is referenced by:  merlem13  1419