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Metamath Proof Explorer

Theorem luklem6

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 luklem6 ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 luk-1 ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
2 luklem5 ⊢ ( ¬ ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) )
3 luklem2 ⊢ ( ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
4 luklem4 ⊢ ( ( ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
5 3 4 luklem1 ⊢ ( ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
6 2 5 luklem1 ⊢ ( ¬ ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
7 luk-1 ⊢ ( ( ¬ ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) )
8 6 7 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
9 luk-1 ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) )
10 8 9 ax-mp ⊢ ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) )
11 luklem4 ⊢ ( ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) )
12 10 11 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) )
13 1 12 luklem1 ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) )