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Theorem exalim 1478
 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 1477. (Contributed by Jim Kingdon, 29-Jul-2018.)
Assertion
Ref Expression
exalim (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem exalim
StepHypRef Expression
1 alnex 1475 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21biimpi 119 . 2 (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
32con2i 616 1 (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329  ∃wex 1468 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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337 This theorem is referenced by:  n0rf  3375  ax9vsep  4051
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