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Metamath Proof Explorer

Theorem luk-1

Description: 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion luk-1 ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 meredith ⊢ ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2 merlem13 ⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )
3 merlem13 ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) → ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) )
4 2 3 ax-mp ⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) )
5 meredith ⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ¬ ¬ ¬ ( 𝜑 → 𝜓 ) → ¬ ( 𝜑 → 𝜓 ) ) ) → ¬ ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) → ( ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
6 4 5 ax-mp ⊢ ( ( ( ( ( ( 𝜒 → 𝜒 ) → ( ¬ ¬ ¬ 𝜑 → ¬ 𝜑 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
7 1 6 ax-mp ⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )