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Theorem alinexa 1539
Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
alinexa (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))

Proof of Theorem alinexa
StepHypRef Expression
1 imnan 659 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21albii 1404 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
3 alnex 1433 . 2 (∀𝑥 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
42, 3bitri 182 1 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295
This theorem is referenced by:  sbnv  1816  ralnex  2369
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