Theorem iineq2

Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	iineq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐵 = 𝐶 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
3		ralbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
5	4	abbidv	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 } = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 } )
6		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 }
7		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 }
8	5 6 7	3eqtr4g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )