Theorem an3andi 1484

Description: Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Assertion

Ref	Expression
an3andi	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)))

Proof of Theorem an3andi

Step	Hyp	Ref	Expression
1		anandi 676	. . 3 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
2		anandi 676	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bianbi 627	. 2 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
4		df-3an 1089	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
5	4	anbi2i 623	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
6		df-3an 1089	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
7	3, 5, 6	3bitr4i 303	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  raltpd  4781