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Theorem alexn 1770
Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
alexn (∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)

Proof of Theorem alexn
StepHypRef Expression
1 exnal 1753 . . 3 (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑)
21albii 1746 . 2 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1705 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3bitri 264 1 (∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1480  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-ex 1704
This theorem is referenced by:  2exnexn  1771  nalset  4793  kmlem2  8970  bj-nalset  32778
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