Theorem anor 981

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 314	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ ¬ (𝜑 ∧ 𝜓))
2		ianor 980	. 2 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	1, 2	xchbinx 333	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 845

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 206  df-an 397  df-or 846

This theorem is referenced by:  pm3.1  990  pm3.11  991  dn1  1056  noran  1533  bropopvvv  8072  swrdnd0  14603  dflim5  42064  ifpananb  42242  iunrelexp0  42438