Theorem adh-minim-ax1

Description: Derivation of ax-1 from adh-minim and ax-mp . Carew Arthur Meredith derived ax-1 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: CpCqp . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minim-ax1	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1	⊢ ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) → ( 𝜓 → 𝜑 ) ) )
2		adh-minim-ax1-ax2-lem1	⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( ( 𝜂 → ( ( 𝜁 → ( 𝜓 → 𝜎 ) ) → ( 𝜁 → 𝜎 ) ) ) → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) )
3		adh-minim-ax1-ax2-lem3	⊢ ( ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ( ( 𝜂 → ( ( 𝜁 → ( 𝜓 → 𝜎 ) ) → ( 𝜁 → 𝜎 ) ) ) → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) )
4		adh-minim-ax1-ax2-lem4	⊢ ( ( ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ( ( 𝜂 → ( ( 𝜁 → ( 𝜓 → 𝜎 ) ) → ( 𝜁 → 𝜎 ) ) ) → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( ( 𝜂 → ( ( 𝜁 → ( 𝜓 → 𝜎 ) ) → ( 𝜁 → 𝜎 ) ) ) → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( ( 𝜂 → ( ( 𝜁 → ( 𝜓 → 𝜎 ) ) → ( 𝜁 → 𝜎 ) ) ) → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) )
6	2 5	ax-mp	⊢ ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) )
7		adh-minim-ax1-ax2-lem4	⊢ ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) ) → ( ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) → ( 𝜓 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
8	6 7	ax-mp	⊢ ( ( 𝜑 → ( ( 𝜓 → ( ( 𝜒 → ( ( 𝜃 → ( 𝜓 → 𝜏 ) ) → ( 𝜃 → 𝜏 ) ) ) → 𝜑 ) ) → ( 𝜓 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
9	1 8	ax-mp	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )