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Theorem ralinexa 3026
 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 437 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21ralbii 3009 . 2 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
3 ralnex 3021 . 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 383  ∀wral 2941  ∃wrex 2942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946  df-rex 2947 This theorem is referenced by:  kmlem7  9016  kmlem13  9022  lspsncv0  19194  ntreq0  20929  lhop1lem  23821  soseq  31879  ltrnnid  35740
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