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Theorem dfexdc 1433
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 1434. (Contributed by Jim Kingdon, 15-Mar-2018.)
Assertion
Ref Expression
dfexdc (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Proof of Theorem dfexdc
StepHypRef Expression
1 alnex 1431 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21a1i 9 . 2 (DECID𝑥𝜑 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
32con2biidc 809 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  DECID wdc 778  wal 1285  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-gen 1381  ax-ie2 1426
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-fal 1293
This theorem is referenced by:  dfrex2dc  2367
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