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Theorem ralinexa 2465
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 680 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21ralbii 2444 . 2 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
3 ralnex 2427 . 2 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
42, 3bitri 183 1 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wral 2417  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-17 1507
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by:  ntreq0  12333
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