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

Theorem dcbii 746
 Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Hypothesis
Ref Expression
dcbii.1 (φψ)
Assertion
Ref Expression
dcbii (DECID φDECID ψ)

Proof of Theorem dcbii
StepHypRef Expression
1 dcbii.1 . . 3 (φψ)
21notbii 593 . . 3 φ ↔ ¬ ψ)
31, 2orbi12i 680 . 2 ((φ ¬ φ) ↔ (ψ ¬ ψ))
4 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
5 df-dc 742 . 2 (DECID ψ ↔ (ψ ¬ ψ))
63, 4, 53bitr4i 201 1 (DECID φDECID ψ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ 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-in1 544  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110  df-dc 742 This theorem is referenced by:  dcbi  843  dcned  2209  euxfr2dc  2720
 Copyright terms: Public domain W3C validator