Theorem anor 509

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		ianor 508	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	bicomi 213	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	2	con2bii 346	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by:  pm3.1  518  pm3.11  519  dn1  1000  3anor  1047  bropopvvv  7142  2wlkonot3v  26402  2spthonot3v  26403  ifpananb  36870  iunrelexp0  37013