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Theorem ralinexa 2991
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
ralinexa (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))

Proof of Theorem ralinexa
StepHypRef Expression
1 imnan 438 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21ralbii 2974 . 2 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
3 ralnex 2986 . 2 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
42, 3bitri 264 1 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wral 2907  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ral 2912  df-rex 2913
This theorem is referenced by:  kmlem7  8922  kmlem13  8928  lspsncv0  19065  ntreq0  20791  lhop1lem  23680  soseq  31449  ltrnnid  34899
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