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Metamath Proof Explorer

Theorem luklem4

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion luklem4 ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )

Proof

Step Hyp Ref Expression
1 luk-2 ⊢ ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
2 luk-2 ⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )
3 luklem3 ⊢ ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) ) )
4 2 3 ax-mp ⊢ ( ( ( ¬ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) )
5 1 4 ax-mp ⊢ ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) )
6 luk-1 ⊢ ( ( ¬ 𝜓 → ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) ) → ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) ) )
7 5 6 ax-mp ⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ¬ 𝜓 → 𝜓 ) )
8 luk-2 ⊢ ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 )
9 7 8 luklem1 ⊢ ( ( ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )