Theorem iineq2d

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011)

	Ref	Expression
Hypotheses	iineq2d.1	⊢ Ⅎ 𝑥 𝜑
	iineq2d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	iineq2d	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iineq2d.1	⊢ Ⅎ 𝑥 𝜑
2		iineq2d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
5		iineq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )
6	4 5	syl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 )