Theorem adh-minim-ax1 47605

Description: Derivation of ax-1 6 from adh-minim 47600 and ax-mp 5. Carew Arthur Meredith derived ax-1 6 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: CpCqp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem adh-minim-ax1

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1 47601	. 2 ⊢ (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜑)) → (𝜓 → 𝜑)))
2		adh-minim-ax1-ax2-lem1 47601	. . . 4 ⊢ ((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 47603	. . . . 5 ⊢ (((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 47604	. . . . 5 ⊢ ((((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 5	. . . 4 ⊢ (((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 5	. . 3 ⊢ ((𝜓 → 𝜑) → (𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜑)))
7		adh-minim-ax1-ax2-lem4 47604	. . 3 ⊢ (((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 5	. 2 ⊢ ((𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜑)) → (𝜓 → 𝜑))) → (𝜑 → (𝜓 → 𝜑)))
9	1, 8	ax-mp 5	1 ⊢ (𝜑 → (𝜓 → 𝜑))

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-ax2-lem5  47606  adh-minim-idALT  47610  adh-minim-pm2.43  47611