Theorem anor 979

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	⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Proof of Theorem anor

Step	Hyp	Ref	Expression
1		notnotb 317	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ ¬ (𝜑 ∧ 𝜓))
2		ianor 978	. 2 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	1, 2	xchbinx 336	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ 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 209  df-an 399  df-or 844

This theorem is referenced by:  pm3.1  988  pm3.11  989  dn1  1052  noran  1526  bropopvvv  7787  swrdnd0  14021  ifpananb  39879  iunrelexp0  40054