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

Theorem dfexdc 1371
 Description: Defining ∃xφ given decidability. It is common in classical logic to define ∃xφ as ¬ ∀x¬ φ but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1372. (Contributed by Jim Kingdon, 15-Mar-2018.)
Assertion
Ref Expression
dfexdc (DECID xφ → (xφ ↔ ¬ x ¬ φ))

Proof of Theorem dfexdc
StepHypRef Expression
1 alnex 1369 . . 3 (x ¬ φ ↔ ¬ xφ)
21a1i 9 . 2 (DECID xφ → (x ¬ φ ↔ ¬ xφ))
32con2biidc 766 1 (DECID xφ → (xφ ↔ ¬ x ¬ φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 733  ∀wal 1226  ∃wex 1362 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-gen 1318  ax-ie2 1364 This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator