Theorem rabeqel

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

	Ref	Expression
Hypotheses	rabeqel.1	$$⊢B=\left\{x\in A|\phi \right\}$$
	rabeqel.2	$$⊢x=C\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rabeqel	$$⊢C\in B\leftrightarrow \left(\psi \wedge C\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rabeqel.1	$$⊢B=\left\{x\in A|\phi \right\}$$
2		rabeqel.2	$$⊢x=C\to \left(\phi \leftrightarrow \psi \right)$$
3	2 1	elrab2	$$⊢C\in B\leftrightarrow \left(C\in A\wedge \psi \right)$$
4	3	biancomi	$$⊢C\in B\leftrightarrow \left(\psi \wedge C\in A\right)$$