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Theorem alinexa 1508
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 632 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21albii 1373 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
3 alnex 1402 . 2 (∀𝑥 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
42, 3bitri 177 1 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wal 1255  wex 1395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-5 1350  ax-gen 1352  ax-ie2 1397
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-fal 1263
This theorem is referenced by:  sbnv  1782  ralnex  2331
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