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 36472
Description: Curry's axiom curryax 892 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 202 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 961 via its inference form jaoi 856; the introduction axioms olc 867 and orc 866 are not needed). Note that this theorem shows that actually, the standard instance of curryax 892 implies the standard instance of peirce 202, which is not the case for the converse bj-peircecurry 36473. (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 856 1 ((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846
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 207  df-or 847
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator