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

Theorem anor 978
 Description: Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.)
Assertion
Ref Expression
anor ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Proof of Theorem anor
StepHypRef Expression
1 notnotb 316 . 2 ((𝜑𝜓) ↔ ¬ ¬ (𝜑𝜓))
2 ianor 977 . 2 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
31, 2xchbinx 335 1 ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844 This theorem is referenced by:  pm3.1  987  pm3.11  988  dn1  1051  noran  1519  bropopvvv  7776  swrdnd0  14009  ifpananb  39737  iunrelexp0  39912
 Copyright terms: Public domain W3C validator