Theorem eqrabi

Description: Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021)

	Ref	Expression
Hypothesis	eqrabi.1	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
Assertion	eqrabi	⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		eqrabi.1	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
2	1	eqabi	⊢ 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
3		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
4	2 3	eqtr4i	⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }