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Theorem peirceroll 85
Description: Over minimal implicational calculus, Peirce's axiom peirce 193 implies an axiom sometimes called "Roll", (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)), of which looinv 194 is a special instance. The converse also holds: substitute (𝜑𝜓) for 𝜒 in Roll and use id 22 and ax-mp 5. (Contributed by BJ, 15-Jun-2021.)
Assertion
Ref Expression
peirceroll ((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)))

Proof of Theorem peirceroll
StepHypRef Expression
1 imim1 83 . 2 (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → ((𝜑𝜓) → 𝜑)))
2 imim2 58 . 2 ((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜒𝜑) → ((𝜑𝜓) → 𝜑)) → ((𝜒𝜑) → 𝜑)))
31, 2syl5 34 1 ((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)))
Colors of variables: wff setvar class
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  32843
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