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

Theorem alinexa 1846
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 1845 . 2 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
2 alnex 1784 . 2 (∀𝑥 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
31, 2bitri 275 1 (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783
This theorem is referenced by:  exnalimn  1847  equsexvw  2009  sbn  2277  ceqsex  3524  ceqsexv  3526  zfregs2  9728  ac6n  10480  nnunb  12468  alexsubALTlem3  23553  nmobndseqi  30032  bj-equsexvwd  35659  difunieq  36255  frege124d  42512  zfregs2VD  43602
  Copyright terms: Public domain W3C validator