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Theorem nrexralim 3190
 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 3189 . . 3 (∃𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
21ralbii 3097 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓))
3 ralnex 3163 . 2 (∀𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓) ↔ ¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
42, 3bitr2i 279 1 (¬ ∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴𝑦𝐵 (𝜑 ∧ ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wral 3070  ∃wrex 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3075  df-rex 3076 This theorem is referenced by: (None)
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