Theorem iniin1

Description: Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	iniin1	⊢ ( 𝐴 ≠ ∅ → ( ∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 ) = ∩ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		iinin1	⊢ ( 𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) = ( ∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 ) )
2	1	eqcomd	⊢ ( 𝐴 ≠ ∅ → ( ∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 ) = ∩ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) )