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Theorem alexnim 1659
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 1657 . . 3 (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
21alimi 1466 . 2 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1510 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3sylib 122 1 (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1362  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472
This theorem is referenced by:  nalset  4155  bj-nalset  15311
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