Theorem rabimbieq

Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021)

	Ref	Expression
Hypotheses	rabimbieq.1	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
	rabimbieq.2	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rabimbieq	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		rabimbieq.1	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
2		rabimbieq.2	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	rabbiia	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 }
4	1 3	eqtri	⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }