Theorem adh-minim-ax1 44361

Description: Derivation of ax-1 6 from adh-minim 44356 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 44357	. 2 ⊢ (𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜑)) → (𝜓 → 𝜑)))
2		adh-minim-ax1-ax2-lem1 44357	. . . 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 44359	. . . . 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 44360	. . . . 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 44360	. . 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  44362  adh-minim-idALT  44366  adh-minim-pm2.43  44367