Theorem 3orit 31596

Description: Closed form of 3ori 1388. (Contributed by Scott Fenton, 20-Apr-2011.)

Assertion

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

Proof of Theorem 3orit

Step	Hyp	Ref	Expression
1		df-3or 1038	. 2 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
2		df-or 385	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) → 𝜒))
3		ioran 511	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
4	3	imbi1i 339	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
5	1, 2, 4	3bitri 286	1 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036

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

This theorem is referenced by: (None)