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Theorem adh-minim-ax2 46305
Description: Derivation of ax-2 7 from adh-minim 46296 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.)
Assertion
Ref Expression
adh-minim-ax2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem adh-minim-ax2
StepHypRef Expression
1 adh-minim-ax2c 46304 . . 3 ((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
2 adh-minim-ax1-ax2-lem3 46299 . . 3 (((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))) → ((𝜑 → (𝜓𝜒)) → ((((𝜃𝜏) → 𝜂) → ((𝜏 → (𝜂𝜁)) → (𝜏𝜁))) → ((𝜑𝜓) → (𝜑𝜒)))))
31, 2ax-mp 5 . 2 ((𝜑 → (𝜓𝜒)) → ((((𝜃𝜏) → 𝜂) → ((𝜏 → (𝜂𝜁)) → (𝜏𝜁))) → ((𝜑𝜓) → (𝜑𝜒))))
4 adh-minim-ax2-lem6 46303 . 2 (((𝜑 → (𝜓𝜒)) → ((((𝜃𝜏) → 𝜂) → ((𝜏 → (𝜂𝜁)) → (𝜏𝜁))) → ((𝜑𝜓) → (𝜑𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒))))
53, 4ax-mp 5 1 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
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  46306  adh-minim-pm2.43  46307
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