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Theorem exnalim 1537
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 1536 . 2 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
21con2i 557 1 (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1241  wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350
This theorem is referenced by:  exanaliim  1538  alexnim  1539  dtru  4254  brprcneu  5134
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