Theorem an3andi 1513

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 689	. . 3 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
2		anandi 689	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
3	1, 2	bianbi 639	. 2 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
4		df-3an 1105	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
5	4	anbi2i 635	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
6		df-3an 1105	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
7	3, 5, 6	3bitr4i 306	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)))

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

This theorem is used by:  raltpd  4749