Theorem elintd

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021)

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

Proof

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