Theorem 3anor 1054

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)

Assertion

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

Proof of Theorem 3anor

Step	Hyp	Ref	Expression
1		df-3an 1039	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2		anor 510	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
3		ianor 509	. . . . 5 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
4	3	orbi1i 542	. . . 4 ⊢ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
5	2, 4	xchbinx 324	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
6		df-3or 1038	. . 3 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
7	5, 6	xchbinxr 325	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
8	1, 7	bitri 264	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037

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 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039

This theorem is referenced by:  3ianor  1055  ne3anior  2887