Theorem intexrab

Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003)

	Ref	Expression
Assertion	intexrab	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∩ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		intexab	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∈ V )
2		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
4	3	inteqi	⊢ ∩ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
5	4	eleq1i	⊢ ( ∩ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V ↔ ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∈ V )
6	1 2 5	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∩ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )