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Theorem dfral2 2976
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2977. (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
dfral2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)

Proof of Theorem dfral2
StepHypRef Expression
1 notnotb 302 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21ralbii 2962 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 ¬ ¬ 𝜑)
3 ralnex 2974 . 2 (∀𝑥𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
42, 3bitri 262 1 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wral 2895  wrex 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-ral 2900  df-rex 2901
This theorem is referenced by:  rexnal  2977  boxcutc  7815  infssuni  8118  ac6n  9168  indstr  11591  trfil3  21450  tglowdim2ln  25292  nmobndseqi  26852  stri  28334  hstri  28342  bnj1204  30168  nosepon  30900  poimirlem1  32404
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