Description: Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rblem1.1 | ⊢ ( ¬ 𝜑 ∨ 𝜓 ) | |
| rblem1.2 | ⊢ ( ¬ 𝜒 ∨ 𝜃 ) | ||
| Assertion | rblem1 | ⊢ ( ¬ ( 𝜑 ∨ 𝜒 ) ∨ ( 𝜓 ∨ 𝜃 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rblem1.1 | ⊢ ( ¬ 𝜑 ∨ 𝜓 ) | |
| 2 | rblem1.2 | ⊢ ( ¬ 𝜒 ∨ 𝜃 ) | |
| 3 | rb-ax1 | ⊢ ( ¬ ( ¬ 𝜒 ∨ 𝜃 ) ∨ ( ¬ ( 𝜓 ∨ 𝜒 ) ∨ ( 𝜓 ∨ 𝜃 ) ) ) | |
| 4 | 2 3 | anmp | ⊢ ( ¬ ( 𝜓 ∨ 𝜒 ) ∨ ( 𝜓 ∨ 𝜃 ) ) |
| 5 | rb-ax2 | ⊢ ( ¬ ( 𝜒 ∨ 𝜓 ) ∨ ( 𝜓 ∨ 𝜒 ) ) | |
| 6 | rb-ax1 | ⊢ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( ¬ ( 𝜒 ∨ 𝜑 ) ∨ ( 𝜒 ∨ 𝜓 ) ) ) | |
| 7 | 1 6 | anmp | ⊢ ( ¬ ( 𝜒 ∨ 𝜑 ) ∨ ( 𝜒 ∨ 𝜓 ) ) |
| 8 | rb-ax2 | ⊢ ( ¬ ( 𝜑 ∨ 𝜒 ) ∨ ( 𝜒 ∨ 𝜑 ) ) | |
| 9 | 7 8 | rbsyl | ⊢ ( ¬ ( 𝜑 ∨ 𝜒 ) ∨ ( 𝜒 ∨ 𝜓 ) ) |
| 10 | 5 9 | rbsyl | ⊢ ( ¬ ( 𝜑 ∨ 𝜒 ) ∨ ( 𝜓 ∨ 𝜒 ) ) |
| 11 | 4 10 | rbsyl | ⊢ ( ¬ ( 𝜑 ∨ 𝜒 ) ∨ ( 𝜓 ∨ 𝜃 ) ) |