Theorem an3andi

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

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

Proof

Step	Hyp	Ref	Expression
1		df-3an	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
2	1	anbi2i	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( 𝜑 ∧ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) )
3		anandi	⊢ ( ( 𝜑 ∧ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
4		anandi	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) )
5	4	anbi1i	⊢ ( ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
6	2 3 5	3bitri	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
7		df-3an	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
8	6 7	bitr4i	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜃 ) ) )