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

Theorem bj-peircecurry 34001
Description: Peirce's axiom peirce 205 implies Curry's axiom curryax 891 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 865 and orc 864; the elimination axiom jao 958 is not needed). See bj-currypeirce 34000 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-peircecurry (𝜑 ∨ (𝜑𝜓))

Proof of Theorem bj-peircecurry
StepHypRef Expression
1 orc 864 . 2 (𝜑 → (𝜑 ∨ (𝜑𝜓)))
2 olc 865 . . 3 ((𝜑𝜓) → (𝜑 ∨ (𝜑𝜓)))
3 peirce 205 . . . 4 ((((𝜑 ∨ (𝜑𝜓)) → 𝜑) → (𝜑 ∨ (𝜑𝜓))) → (𝜑 ∨ (𝜑𝜓)))
4 peirce 205 . . . . 5 (((𝜑𝜓) → 𝜑) → 𝜑)
5 peirceroll 85 . . . . 5 ((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜑𝜓) → (𝜑 ∨ (𝜑𝜓))) → (((𝜑 ∨ (𝜑𝜓)) → 𝜑) → 𝜑)))
64, 5ax-mp 5 . . . 4 (((𝜑𝜓) → (𝜑 ∨ (𝜑𝜓))) → (((𝜑 ∨ (𝜑𝜓)) → 𝜑) → 𝜑))
7 peirceroll 85 . . . 4 (((((𝜑 ∨ (𝜑𝜓)) → 𝜑) → (𝜑 ∨ (𝜑𝜓))) → (𝜑 ∨ (𝜑𝜓))) → ((((𝜑 ∨ (𝜑𝜓)) → 𝜑) → 𝜑) → ((𝜑 → (𝜑 ∨ (𝜑𝜓))) → (𝜑 ∨ (𝜑𝜓)))))
83, 6, 7mpsyl 68 . . 3 (((𝜑𝜓) → (𝜑 ∨ (𝜑𝜓))) → ((𝜑 → (𝜑 ∨ (𝜑𝜓))) → (𝜑 ∨ (𝜑𝜓))))
92, 8ax-mp 5 . 2 ((𝜑 → (𝜑 ∨ (𝜑𝜓))) → (𝜑 ∨ (𝜑𝜓)))
101, 9ax-mp 5 1 (𝜑 ∨ (𝜑𝜓))
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)
  Copyright terms: Public domain W3C validator