Theorem 3ioran 1123

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 999	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2	1	anbi1i 636	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3		ioran 999	. . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
4		df-3or 1104	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	3, 4	xchnxbir 336	. 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒))
6		df-3an 1105	. 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:  3oran  1126  cadnot  1648  fvtp0  7206  lcmftp  16811  prm23ge5  16993  cnfldfun  21692  fbunfip  24188  frgrregord013  30996  wl-nfeqfb  38468  usgrexmpl2trifr  49134  gpg5nbgrvtx03starlem1  49165  gpg5nbgrvtx03starlem2  49166  gpg5nbgrvtx03starlem3  49167  gpg5nbgrvtx13starlem1  49168  gpg5nbgrvtx13starlem2  49169  gpg5nbgrvtx13starlem3  49170  gpg5edgnedg  49227