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Theorem exnalim 1626
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 1625 . 2 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
21con2i 617 1 (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1330  wex 1469
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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438
This theorem is referenced by:  exanaliim  1627  alexnim  1628  dtru  4482  brprcneu  5421  bj-nnal  13118
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