Theorem 3ianor 1124

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 336. (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 997	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	orbi1i 927	. 2 ⊢ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
3		ianor 997	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
4		df-3an 1105	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
5	3, 4	xchnxbir 336	. 2 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
6		df-3or 1104	. 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
7	2, 5, 6	3bitr4i 306	1 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105

This theorem is used by:  3anor  1125  tppreqb  4768  otthne  5462  fr3nr  7773  bropopvvv  8089  prinfzo0  13776  elfznelfzo  13851  ssnn0fi  14071  hashtpg  14572  hash3tpde  14580  swrdnd0  14749  pfxnd0  14780  lcmfunsnlem2lem2  16751  prm23ge5  16929  2irrexpq  26997  lpni  30990  xrdifh  33280  dvasin  38467  dflim5  44184  limcicciooub  46479  2zrngnring  49187