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Theorem exanaliim 1579
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 816 . . 3 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
21eximi 1532 . 2 (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑𝜓))
3 exnalim 1578 . 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 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by:  rexnalim  2364  nssr  3068  nssssr  4012  brprcneu  5244
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