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Theorem alinexa 1767
 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 1766 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
2 alnex 1703 . 2 (∀𝑥 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
31, 2bitri 264 1 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  equs3  1872  ralnexOLD  2989  r2exlem  3054  zfregs2  8569  ac6n  9267  nnunb  11248  alexsubALTlem3  21793  nmobndseqi  27522  bj-exnalimn  32305  bj-ssbn  32336  frege124d  37573  zfregs2VD  38598
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