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Theorem dfral2 2977
 Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2978. (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
dfral2 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)

Proof of Theorem dfral2
StepHypRef Expression
1 notnotb 303 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21ralbii 2963 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 ¬ ¬ 𝜑)
3 ralnex 2975 . 2 (∀𝑥𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
42, 3bitri 263 1 (∀𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wral 2896  ∃wrex 2897 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-an 385  df-ex 1696  df-ral 2901  df-rex 2902 This theorem is referenced by:  rexnal  2978  boxcutc  7837  infssuni  8140  ac6n  9190  indstr  11632  trfil3  21502  tglowdim2ln  25346  nmobndseqi  27018  stri  28500  hstri  28508  bnj1204  30334  nosepon  31066  poimirlem1  32580
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