Theorem andi3or 41632

Description: Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)

Assertion

Ref	Expression
andi3or	⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))

Proof of Theorem andi3or

Step	Hyp	Ref	Expression
1		andi 1005	. . 3 ⊢ ((𝜑 ∧ ((𝜓 ∨ 𝜒) ∨ 𝜃)) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜑 ∧ 𝜃)))
2		andi 1005	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
3	2	orbi1i 911	. . 3 ⊢ (((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜑 ∧ 𝜃)))
4	1, 3	bitri 274	. 2 ⊢ ((𝜑 ∧ ((𝜓 ∨ 𝜒) ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜑 ∧ 𝜃)))
5		df-3or 1087	. . 3 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))
6	5	anbi2i 623	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∨ 𝜒) ∨ 𝜃)))
7		df-3or 1087	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜑 ∧ 𝜃)))
8	4, 6, 7	3bitr4i 303	1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085

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 206  df-an 397  df-or 845  df-3or 1087

This theorem is referenced by:  uneqsn  41633