Description: luk-3 derived from Russell-Bernays'.
This theorem, along with re1axmp , re2luk1 , and re2luk2 shows that rb-ax1 , rb-ax2 , rb-ax3 , and rb-ax4 , along with anmp , can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | re2luk3 | ⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rb-imdf | ⊢ ¬ ( ¬ ( ¬ ( ¬ 𝜑 → 𝜓 ) ∨ ( ¬ ¬ 𝜑 ∨ 𝜓 ) ) ∨ ¬ ( ¬ ( ¬ ¬ 𝜑 ∨ 𝜓 ) ∨ ( ¬ 𝜑 → 𝜓 ) ) ) | |
| 2 | 1 | rblem7 | ⊢ ( ¬ ( ¬ ¬ 𝜑 ∨ 𝜓 ) ∨ ( ¬ 𝜑 → 𝜓 ) ) |
| 3 | rb-ax4 | ⊢ ( ¬ ( ¬ 𝜑 ∨ ¬ 𝜑 ) ∨ ¬ 𝜑 ) | |
| 4 | rb-ax3 | ⊢ ( ¬ ¬ 𝜑 ∨ ( ¬ 𝜑 ∨ ¬ 𝜑 ) ) | |
| 5 | 3 4 | rbsyl | ⊢ ( ¬ ¬ 𝜑 ∨ ¬ 𝜑 ) |
| 6 | rb-ax2 | ⊢ ( ¬ ( ¬ ¬ 𝜑 ∨ ¬ 𝜑 ) ∨ ( ¬ 𝜑 ∨ ¬ ¬ 𝜑 ) ) | |
| 7 | 5 6 | anmp | ⊢ ( ¬ 𝜑 ∨ ¬ ¬ 𝜑 ) |
| 8 | rblem2 | ⊢ ( ¬ ( ¬ 𝜑 ∨ ¬ ¬ 𝜑 ) ∨ ( ¬ 𝜑 ∨ ( ¬ ¬ 𝜑 ∨ 𝜓 ) ) ) | |
| 9 | 7 8 | anmp | ⊢ ( ¬ 𝜑 ∨ ( ¬ ¬ 𝜑 ∨ 𝜓 ) ) |
| 10 | 2 9 | rbsyl | ⊢ ( ¬ 𝜑 ∨ ( ¬ 𝜑 → 𝜓 ) ) |
| 11 | rb-imdf | ⊢ ¬ ( ¬ ( ¬ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ∨ ( ¬ 𝜑 ∨ ( ¬ 𝜑 → 𝜓 ) ) ) ∨ ¬ ( ¬ ( ¬ 𝜑 ∨ ( ¬ 𝜑 → 𝜓 ) ) ∨ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ) ) | |
| 12 | 11 | rblem7 | ⊢ ( ¬ ( ¬ 𝜑 ∨ ( ¬ 𝜑 → 𝜓 ) ) ∨ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) ) |
| 13 | 10 12 | anmp | ⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜓 ) ) |