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Theorem dfexdc 1390
 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 1391. (Contributed by Jim Kingdon, 15-Mar-2018.)
Assertion
Ref Expression
dfexdc (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Proof of Theorem dfexdc
StepHypRef Expression
1 alnex 1388 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21a1i 9 . 2 (DECID𝑥𝜑 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
32con2biidc 773 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742  ∀wal 1241  ∃wex 1381 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 630  ax-5 1336  ax-gen 1338  ax-ie2 1383 This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249 This theorem is referenced by: (None)
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