Theorem an3andi 1478

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 674	. . 3 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
2		anandi 674	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bianbi 625	. 2 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
4		df-3an 1086	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
5	4	anbi2i 621	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
6		df-3an 1086	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
7	3, 5, 6	3bitr4i 302	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1084

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 395  df-3an 1086

This theorem is referenced by:  raltpd  4790