Theorem adh-minim-ax2

Description: Derivation of ax-2 from adh-minim and ax-mp . Carew Arthur Meredith derived ax-2 inA 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
Assertion	adh-minim-ax2	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax2c	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) )
2		adh-minim-ax1-ax2-lem3	⊢ ( ( ( 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	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( ( 𝜃 → 𝜏 ) → 𝜂 ) → ( ( 𝜏 → ( 𝜂 → 𝜁 ) ) → ( 𝜏 → 𝜁 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) )
4		adh-minim-ax2-lem6	⊢ ( ( ( 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	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )