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Theorem nrexralim 3145
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 3144 . . 3 (∃𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
21ralbii 3127 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓))
3 ralnex 3139 . 2 (∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓) ↔ ¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
42, 3bitr2i 267 1 (¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wral 3055  wrex 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-ral 3060  df-rex 3061
This theorem is referenced by: (None)
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