Theorem iineq2i

Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Hypothesis	iuneq2i.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion	iineq2i	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		iuneq2i.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
2		iineq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )
3	2 1	mprg	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶