Theorem adh-minim-ax1 48020

Description: Derivation of ax-1 6 from adh-minim 48015 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 48016	. 2 ⊢ (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜑)) → (𝜓 → 𝜑)))
2		adh-minim-ax1-ax2-lem1 48016	. . . 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 48018	. . . . 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 48019	. . . . 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 48019	. . 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))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  adh-minim-ax2-lem5  48021  adh-minim-idALT  48025  adh-minim-pm2.43  48026