Theorem adh-minim-ax2 43327

Description: Derivation of ax-2 7 from adh-minim 43318 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 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Proof of Theorem adh-minim-ax2

Step	Hyp	Ref	Expression
1		adh-minim-ax2c 43326	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
2		adh-minim-ax1-ax2-lem3 43321	. . 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 43325	. 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)))

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  43328  adh-minim-pm2.43  43329