Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | tbwlem5 | ⊢ ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tbw-ax2 | ⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) ) | |
| 2 | tbw-ax1 | ⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) ) | |
| 3 | 1 2 | tbwsyl | ⊢ ( 𝜑 → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) ) |
| 4 | tbwlem1 | ⊢ ( ( 𝜑 → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) ) → ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) ) ) | |
| 5 | 3 4 | ax-mp | ⊢ ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) ) |
| 6 | tbwlem4 | ⊢ ( ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 ) ) | |
| 7 | 5 6 | ax-mp | ⊢ ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 ) |