Description: Derivation of pm2.43 (also called "hilbert" or W) from ax-mp and minimp . It uses the classical derivation from ax-1 and ax-2 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 (which only requires ax-mp ). (Contributed by BJ, 31-May-2021) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | minimp-pm2.43 | ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | minimp-ax2 | ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) | |
| 2 | minimp-ax1 | ⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) | |
| 3 | minimp-ax2 | ⊢ ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) ) | |
| 4 | 2 3 | ax-mp | ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) |
| 5 | minimp-ax2 | ⊢ ( ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) ) | |
| 6 | 1 4 5 | mp2 | ⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) |