Theorem 3ianor 1106

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.) Shorten with xchnxbir 333. (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 983	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	orbi1i 913	. 2 ⊢ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
3		ianor 983	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
4		df-3an 1088	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
5	3, 4	xchnxbir 333	. 2 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
6		df-3or 1087	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
7	2, 5, 6	3bitr4i 303	1 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086

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 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088

This theorem is referenced by:  3anor  1107  tppreqb  4786  otthne  5466  fr3nr  7771  bropopvvv  8094  prinfzo0  13720  elfznelfzo  13793  ssnn0fi  14008  hashtpg  14508  hash3tpde  14516  swrdnd0  14680  pfxnd0  14711  lcmfunsnlem2lem2  16663  prm23ge5  16840  2irrexpq  26697  lpni  30466  xrdifh  32762  dvasin  37733  dflim5  43320  limcicciooub  45633  2zrngnring  48200