Description: Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | merlem8 | ⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meredith | ⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) | |
2 | merlem7 | ⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝜑 ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) ) | |
3 | 1 2 | ax-mp | ⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( ( ( 𝜒 → 𝜏 ) → ( ¬ 𝜃 → ¬ 𝜓 ) ) → 𝜃 ) ) |