Theorem iineq1d

Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypothesis	iineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	iineq1d	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		iineq1	⊢ ( 𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )