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Theorem alinexa 1843
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 imnang 1842 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
2 alnex 1781 . 2 (∀𝑥 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
31, 2bitri 274 1 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1537  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780
This theorem is referenced by:  exnalimn  1844  equsexvw  2006  sbn  2274  ceqsex  3522  ceqsexv  3524  zfregs2  9730  ac6n  10482  nnunb  12472  alexsubALTlem3  23773  nmobndseqi  30299  bj-equsexvwd  35962  difunieq  36558  frege124d  42814  zfregs2VD  43904
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