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  4754  otthne  5424  fr3nr  7705  bropopvvv  8020  prinfzo0  13598  elfznelfzo  13673  ssnn0fi  13892  hashtpg  14392  hash3tpde  14400  swrdnd0  14565  pfxnd0  14596  lcmfunsnlem2lem2  16550  prm23ge5  16727  2irrexpq  26667  lpni  30460  xrdifh  32763  dvasin  37754  dflim5  43432  limcicciooub  45745  2zrngnring  48368