Theorem elintab

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993)

	Ref	Expression
Hypothesis	elintab.ex	⊢ 𝐴 ∈ V
Assertion	elintab	⊢ ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		elintab.ex	⊢ 𝐴 ∈ V
2		elintabg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) )