Theorem or3di 30225

Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Assertion

Ref	Expression
or3di	⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))

Proof of Theorem or3di

Step	Hyp	Ref	Expression
1		df-3an 1085	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜏))
2	1	orbi2i 909	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ (𝜑 ∨ ((𝜓 ∧ 𝜒) ∧ 𝜏)))
3		ordi 1002	. . 3 ⊢ ((𝜑 ∨ ((𝜓 ∧ 𝜒) ∧ 𝜏)) ↔ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
4		ordi 1002	. . . 4 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	4	anbi1i 625	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∨ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
6	2, 3, 5	3bitri 299	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
7		df-3an 1085	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
8	6, 7	bitr4i 280	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083

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 399  df-or 844  df-3an 1085

This theorem is referenced by:  or3dir  30226