Metamath Proof Explorer


Theorem merlem2

Description: Step 4 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 merlem2 ( ( ( 𝜑𝜑 ) → 𝜒 ) → ( 𝜃𝜒 ) )

Proof

Step Hyp Ref Expression
1 merlem1 ( ( ( ( 𝜒𝜒 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → ( 𝜑𝜑 ) )
2 meredith ( ( ( ( ( 𝜒𝜒 ) → ( ¬ 𝜑 → ¬ 𝜃 ) ) → 𝜑 ) → ( 𝜑𝜑 ) ) → ( ( ( 𝜑𝜑 ) → 𝜒 ) → ( 𝜃𝜒 ) ) )
3 1 2 ax-mp ( ( ( 𝜑𝜑 ) → 𝜒 ) → ( 𝜃𝜒 ) )