Theorem iineq1

Description: Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998)

	Ref	Expression
Assertion	iineq1	⊢ ( 𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		raleq	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ) )
2	1	abbidv	⊢ ( 𝐴 = 𝐵 → { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 } = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 } )
3		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 }
4		df-iin	⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 }
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )