Theorem or3di 33050

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 1105	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜏))
2	1	orbi2i 926	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ (𝜑 ∨ ((𝜓 ∧ 𝜒) ∧ 𝜏)))
3		ordi 1023	. . 3 ⊢ ((𝜑 ∨ ((𝜓 ∧ 𝜒) ∧ 𝜏)) ↔ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
4		ordi 1023	. . . 4 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	4	anbi1i 636	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∨ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
6	2, 3, 5	3bitri 300	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
7		df-3an 1105	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜑 ∨ 𝜏)))
8	6, 7	bitr4i 281	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ 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-3an 1105

This theorem is used by:  or3dir  33051