Theorem iineq1i

Description: Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	iineq1i.1	⊢ 𝐴 = 𝐵
Assertion	iineq1i	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶

Proof

Step	Hyp	Ref	Expression
1		iineq1i.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	imbi1i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶 ) )
4	3	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 )
5	4	abbii	⊢ { 𝑡 ∣ ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 } = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
6		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 }
7		df-iin	⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
8	5 6 7	3eqtr4i	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶