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Theorem bj-currypeirce 33966
Description: Curry's axiom curryax 891 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 205 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 958 via its inference form jaoi 854; the introduction axioms olc 865 and orc 864 are not needed). Note that this theorem shows that actually, the standard instance of curryax 891 implies the standard instance of peirce 205, which is not the case for the converse bj-peircecurry 33967. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-currypeirce ((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))

Proof of Theorem bj-currypeirce
StepHypRef Expression
1 ax-1 6 . 2 (𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
2 pm2.27 42 . 2 ((𝜑𝜓) → (((𝜑𝜓) → 𝜑) → 𝜑))
31, 2jaoi 854 1 ((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-or 845
This theorem is referenced by: (None)
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