Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-currypeirce Structured version   Visualization version   GIF version

Theorem bj-currypeirce 34733
Description: Curry's axiom curryax 891 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 201 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 201, which is not the case for the converse bj-peircecurry 34734. (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 206  df-or 845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator