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Theorem exalim 1495
Description: One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1494. (Contributed by Jim Kingdon, 29-Jul-2018.)
Assertion
Ref Expression
exalim (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem exalim
StepHypRef Expression
1 alnex 1492 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21biimpi 119 . 2 (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
32con2i 622 1 (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1346  wex 1485
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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354
This theorem is referenced by:  n0rf  3427  ax9vsep  4112
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