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Theorem alexnim 1612
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 1610 . . 3 (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
21alimi 1416 . 2 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1460 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3sylib 121 1 (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422
This theorem is referenced by:  nalset  4028  bj-nalset  13020
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