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  4769  otthne  5446  fr3nr  7748  bropopvvv  8069  prinfzo0  13659  elfznelfzo  13733  ssnn0fi  13950  hashtpg  14450  hash3tpde  14458  swrdnd0  14622  pfxnd0  14653  lcmfunsnlem2lem2  16609  prm23ge5  16786  2irrexpq  26640  lpni  30409  xrdifh  32703  dvasin  37698  dflim5  43318  limcicciooub  45635  2zrngnring  48243