Theorem merlem1 1641

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem1	⊢ (((𝜒 → (¬ 𝜑 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))

Proof of Theorem merlem1

Step	Hyp	Ref	Expression
1		meredith 1640	. . 3 ⊢ (((((¬ 𝜑 → 𝜓) → (¬ (¬ 𝜏 → ¬ 𝜒) → ¬ ¬ (¬ 𝜑 → 𝜓))) → (¬ 𝜏 → ¬ 𝜒)) → 𝜏) → ((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑 → 𝜓) → ¬ 𝜑)))
2		meredith 1640	. . 3 ⊢ ((((((¬ 𝜑 → 𝜓) → (¬ (¬ 𝜏 → ¬ 𝜒) → ¬ ¬ (¬ 𝜑 → 𝜓))) → (¬ 𝜏 → ¬ 𝜒)) → 𝜏) → ((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑 → 𝜓) → ¬ 𝜑))) → ((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑 → 𝜓) → ¬ 𝜑)) → (¬ 𝜑 → 𝜓)) → (𝜒 → (¬ 𝜑 → 𝜓))))
3	1, 2	ax-mp 5	. 2 ⊢ ((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑 → 𝜓) → ¬ 𝜑)) → (¬ 𝜑 → 𝜓)) → (𝜒 → (¬ 𝜑 → 𝜓)))
4		meredith 1640	. 2 ⊢ (((((𝜏 → ¬ 𝜑) → (¬ (¬ 𝜑 → 𝜓) → ¬ 𝜑)) → (¬ 𝜑 → 𝜓)) → (𝜒 → (¬ 𝜑 → 𝜓))) → (((𝜒 → (¬ 𝜑 → 𝜓)) → 𝜏) → (𝜑 → 𝜏)))
5	3, 4	ax-mp 5	1 ⊢ (((𝜒 → (¬ 𝜑 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  merlem2  1642  merlem5  1645  luk-3  1656