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Theorem alexnim 1539
 Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
alexnim (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)

Proof of Theorem alexnim
StepHypRef Expression
1 exnalim 1537 . . 3 (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
21alimi 1344 . 2 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1388 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3sylib 127 1 (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
 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:  nalset  3887  bj-nalset  10015
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