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

Theorem looinvdc 821
 Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 794, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
looinvdc (DECID 𝜑 → (((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑)))

Proof of Theorem looinvdc
StepHypRef Expression
1 imim1 70 . 2 (((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → ((𝜑𝜓) → 𝜑)))
2 peircedc 820 . 2 (DECID 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
31, 2syl9r 67 1 (DECID 𝜑 → (((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4  DECID wdc 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator