Theorem 3ioran 993

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 752	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2	1	anbi1i 458	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3		ioran 752	. . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
4		df-3or 979	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	3, 4	xchnxbir 681	. 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
6		df-3an 980	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
7	2, 5, 6	3bitr4i 212	1 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∨ w3o 977   ∧ w3a 978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980

This theorem is referenced by:  ne3anior  2435  onntri35  7232