Theorem 3anor 1104

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)

Assertion

Ref	Expression
3anor	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3anor

Step	Hyp	Ref	Expression
1		3ianor 1103	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
2	1	con1bii 359	. 2 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
3	2	bicomi 226	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ w3o 1082   ∧ w3a 1083

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  df-3or 1084  df-3an 1085

This theorem is referenced by:  ne3anior  3110  swrdnd0  14018