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Theorem exnalim 1639
Description: One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
exnalim (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Proof of Theorem exnalim
StepHypRef Expression
1 alexim 1638 . 2 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
21con2i 622 1 (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
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-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454
This theorem is referenced by:  exanaliim  1640  alexnim  1641  nnal  1642  dtru  4542  brprcneu  5487
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