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Metamath Proof Explorer

Theorem merlem1

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.)

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

Proof

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