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Theorem dfexdc 1462
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 1463. (Contributed by Jim Kingdon, 15-Mar-2018.)
Assertion
Ref Expression
dfexdc (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Proof of Theorem dfexdc
StepHypRef Expression
1 alnex 1460 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21a1i 9 . 2 (DECID𝑥𝜑 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
32con2biidc 849 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  DECID wdc 804  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-gen 1410  ax-ie2 1455
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-fal 1322
This theorem is referenced by:  dfrex2dc  2405
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