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		df-3an 1085	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
2	1	anbi2i 624	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
3		anandi 674	. . 3 ⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
4		anandi 674	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
5	4	anbi1i 625	. . 3 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
6	2, 3, 5	3bitri 299	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
7		df-3an 1085	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜃)))
8	6, 7	bitr4i 280	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by:  raltpd  4709