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Theorem alexn 1843
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 1825 . . 3 (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑)
21albii 1817 . 2 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1779 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3bitri 274 1 (∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1535  wex 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807
This theorem depends on definitions:  df-bi 206  df-ex 1778
This theorem is referenced by:  2exnexn  1844  nalset  5240  kmlem2  9935
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