Theorem 3ioran 1106

Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)

Assertion

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

Proof of Theorem 3ioran

Step	Hyp	Ref	Expression
1		ioran 982	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2	1	anbi1i 625	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3		ioran 982	. . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
4		df-3or 1088	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	3, 4	xchnxbir 333	. 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
6		df-3an 1089	. 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 205   ∧ wa 397   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087

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 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089

This theorem is referenced by:  3oran  1109  cadnot  1614  lcmftp  16390  prm23ge5  16565  cnfldfun  20658  fbunfip  23069  frgrregord013  28808  wl-nfeqfb  35743  sn-inelr  40630