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Mirrors > Home > MPE Home > Th. List > Mathboxes > adh-minim-ax2 | Structured version Visualization version GIF version |
Description: Derivation of ax-2 7 from adh-minim 43311 and ax-mp 5. Carew Arthur Meredith derived ax-2 7 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adh-minim-ax2 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adh-minim-ax2c 43319 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | |
2 | adh-minim-ax1-ax2-lem3 43314 | . . 3 ⊢ (((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) |
4 | adh-minim-ax2-lem6 43318 | . 2 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: adh-minim-idALT 43321 adh-minim-pm2.43 43322 |
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