Theorem 3ioran 1119

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 997	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2	1	anbi1i 633	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3		ioran 997	. . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
4		df-3or 1100	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	3, 4	xchnxbir 335	. 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
6		df-3an 1101	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
7	2, 5, 6	3bitr4i 305	1 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∨ w3o 1098   ∧ w3a 1099

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 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101

This theorem is referenced by:  3oran  1122  cadnot  1636  lcmftp  16671  prm23ge5  16852  cnfldfun  21439  fbunfip  23930  frgrregord013  30598  wl-nfeqfb  38040  usgrexmpl2trifr  48660  gpg5nbgrvtx03starlem1  48691  gpg5nbgrvtx03starlem2  48692  gpg5nbgrvtx03starlem3  48693  gpg5nbgrvtx13starlem1  48694  gpg5nbgrvtx13starlem2  48695  gpg5nbgrvtx13starlem3  48696  gpg5edgnedg  48753