Theorem peirceroll 85

Description: Over minimal implicational calculus, Peirce's axiom peirce 201 implies an axiom sometimes called "Roll", (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → 𝜑)), of which looinv 202 is a special instance. The converse also holds: substitute (𝜑 → 𝜓) for 𝜒 in Roll and use id 22 and ax-mp 5. (Contributed by BJ, 15-Jun-2021.)

Assertion

Ref	Expression
peirceroll	⊢ ((((𝜑 → 𝜓) → 𝜑) → 𝜑) → (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → 𝜑)))

Proof of Theorem peirceroll

Step	Hyp	Ref	Expression
1		imim1 83	. 2 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		id 22	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜑) → 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3	1, 2	syl9r 78	1 ⊢ ((((𝜑 → 𝜓) → 𝜑) → 𝜑) → (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → 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:  bj-peircecurry  34665