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 | luklem2 | ⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | luk-1 | ⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( ( ¬ 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) | |
| 2 | luk-3 | ⊢ ( 𝜓 → ( ¬ 𝜓 → 𝜒 ) ) | |
| 3 | luk-1 | ⊢ ( ( 𝜓 → ( ¬ 𝜓 → 𝜒 ) ) → ( ( ( ¬ 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) | |
| 4 | 2 3 | ax-mp | ⊢ ( ( ( ¬ 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |
| 5 | 1 4 | luklem1 | ⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |
| 6 | luk-1 | ⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) | |
| 7 | 5 6 | luklem1 | ⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) |