Theorem impsingle 1627

Description: The shortest single axiom for implicational calculus, due to Lukasiewicz. It is now known to be the unique shortest axiom. The axiom is proved here starting from { ax-1 6, ax-2 7 and peirce 202 }. (Contributed by Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
impsingle	⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → (𝜃 → 𝜑)))

Proof of Theorem impsingle

Step	Hyp	Ref	Expression
1		imim1 83	. 2 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		peirce 202	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
3	2	a1d 25	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜃 → 𝜑))
4	1, 3	syl6 35	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → (𝜃 → 𝜑)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  impsingle-step4  1628  impsingle-step8  1629  impsingle-step15  1631  impsingle-step18  1632  impsingle-step20  1634  impsingle-step22  1636