Theorem eliind2

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	eliind2.1	⊢ Ⅎ 𝑥 𝜑
	eliind2.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	eliind2.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ 𝐶 )
Assertion	eliind2	⊢ ( 𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eliind2.1	⊢ Ⅎ 𝑥 𝜑
2		eliind2.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		eliind2.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ 𝐶 )
4	3	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 → 𝐴 ∈ 𝐶 ) )
5	1 4	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )
6		eliin	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )
7	2 6	syl	⊢ ( 𝜑 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )
8	5 7	mpbird	⊢ ( 𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 )