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Theorem nrexralim 2982
Description: Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
nrexralim (¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))

Proof of Theorem nrexralim
StepHypRef Expression
1 rexanali 2981 . . 3 (∃𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
21ralbii 2963 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓))
3 ralnex 2975 . 2 (∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓) ↔ ¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
42, 3bitr2i 264 1 (¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  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: (None)
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