Theorem 3ianor 1103

Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.)

Assertion

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

Proof of Theorem 3ianor

Step	Hyp	Ref	Expression
1		ianor 978	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	orbi1i 910	. 2 ⊢ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
3		ianor 978	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
4		df-3an 1085	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
5	3, 4	xchnxbir 335	. 2 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
6		df-3or 1084	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
7	2, 5, 6	3bitr4i 305	1 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ 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:  3anor  1104  tppreqb  4731  fr3nr  7488  bropopvvv  7779  prinfzo0  13070  elfznelfzo  13136  ssnn0fi  13347  hashtpg  13837  swrdnd0  14013  pfxnd0  14044  lcmfunsnlem2lem2  15977  prm23ge5  16146  2irrexpq  25307  lpni  28251  xrdifh  30497  dvasin  34972  limcicciooub  41911  2zrngnring  44217