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Metamath Proof Explorer

Theorem merlem10

Description: Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merlem10 ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 meredith ⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) )
2 meredith ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) )
3 merlem9 ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) ) )
4 2 3 ax-mp ⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) ) )
5 1 4 ax-mp ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜃 → ( 𝜑 → 𝜓 ) ) )