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Theorem exanaliim 1538
 Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
exanaliim (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑𝜓))

Proof of Theorem exanaliim
StepHypRef Expression
1 annimim 782 . . 3 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
21eximi 1491 . 2 (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑𝜓))
3 exnalim 1537 . 2 (∃𝑥 ¬ (𝜑𝜓) → ¬ ∀𝑥(𝜑𝜓))
42, 3syl 14 1 (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀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:  rexnalim  2317  nssr  3003  nssssr  3958  brprcneu  5171
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