Theorem elintrabg

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007)

	Ref	Expression
Assertion	elintrabg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ 𝐴 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ) )
2		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) )
3	2	imbi2d	⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 → 𝑦 ∈ 𝑥 ) ↔ ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
4	3	ralbidv	⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
5		vex	⊢ 𝑦 ∈ V
6	5	elintrab	⊢ ( 𝑦 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝑦 ∈ 𝑥 ) )
7	1 4 6	vtoclbg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )