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Theorem alexn 1760
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 1744 . . 3 (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑)
21albii 1737 . 2 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1697 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3bitri 263 1 (∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-ex 1696
This theorem is referenced by:  2exnexn  1761  nalset  4723  kmlem2  8856  bj-nalset  31982
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