 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  peircedc Structured version   GIF version

Theorem peircedc 819
 Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 743, condc 748, or notnotdc 765 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
Assertion
Ref Expression
peircedc (DECID φ → (((φψ) → φ) → φ))

Proof of Theorem peircedc
StepHypRef Expression
1 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
2 ax-1 5 . . 3 (φ → (((φψ) → φ) → φ))
3 pm2.21 547 . . . . 5 φ → (φψ))
43imim1i 54 . . . 4 (((φψ) → φ) → (¬ φφ))
54com12 27 . . 3 φ → (((φψ) → φ) → φ))
62, 5jaoi 635 . 2 ((φ ¬ φ) → (((φψ) → φ) → φ))
71, 6sylbi 114 1 (DECID φ → (((φψ) → φ) → φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 628  DECID wdc 741 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110  df-dc 742 This theorem is referenced by:  looinvdc  820  exmoeudc  1960
 Copyright terms: Public domain W3C validator