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	$$⊢B=\left\{x\in A|\phi \right\}$$
	rabimbieq.2	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rabimbieq	$$⊢B=\left\{x\in A|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabimbieq.1	$$⊢B=\left\{x\in A|\phi \right\}$$
2		rabimbieq.2	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
3	2	rabbiia	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in A|\psi \right\}$$
4	1 3	eqtri	$$⊢B=\left\{x\in A|\psi \right\}$$