Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  alinexa Structured version   Visualization version   GIF version

Theorem alinexa 1920
 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 1919 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
2 alnex 1854 . 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 382  ∀wal 1629  ∃wex 1852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by:  equs3  2044  r2exlem  3207  zfregs2  8773  ac6n  9509  nnunb  11490  alexsubALTlem3  22073  nmobndseqi  27974  bj-exnalimn  32947  bj-ssbn  32978  frege124d  38579  zfregs2VD  39598
 Copyright terms: Public domain W3C validator