Theorem abeqin

Description: Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021)

	Ref	Expression
Hypotheses	abeqin.1	⊢ 𝐴 = ( 𝐵 ∩ 𝐶 )
	abeqin.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
Assertion	abeqin	⊢ 𝐴 = { 𝑥 ∈ 𝐶 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		abeqin.1	⊢ 𝐴 = ( 𝐵 ∩ 𝐶 )
2		abeqin.2	⊢ 𝐵 = { 𝑥 ∣ 𝜑 }
3	2	ineq1i	⊢ ( 𝐵 ∩ 𝐶 ) = ( { 𝑥 ∣ 𝜑 } ∩ 𝐶 )
4		dfrab2	⊢ { 𝑥 ∈ 𝐶 ∣ 𝜑 } = ( { 𝑥 ∣ 𝜑 } ∩ 𝐶 )
5	3 1 4	3eqtr4i	⊢ 𝐴 = { 𝑥 ∈ 𝐶 ∣ 𝜑 }