Theorem rabeqsn

Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019) (Proof shortened by AV, 26-Aug-2022)

	Ref	Expression
Assertion	rabeqsn	$$⊢\left\{x\in V|\phi \right\}=\left\{X\right\}\leftrightarrow \forall x\left(\left(x\in V\wedge \phi \right)\leftrightarrow x=X\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-rab	$$⊢\left\{x\in V|\phi \right\}=\left\{x|\left(x\in V\wedge \phi \right)\right\}$$
2	1	eqeq1i	$$⊢\left\{x\in V|\phi \right\}=\left\{X\right\}\leftrightarrow \left\{x|\left(x\in V\wedge \phi \right)\right\}=\left\{X\right\}$$
3		absn	$$⊢\left\{x|\left(x\in V\wedge \phi \right)\right\}=\left\{X\right\}\leftrightarrow \forall x\left(\left(x\in V\wedge \phi \right)\leftrightarrow x=X\right)$$
4	2 3	bitri	$$⊢\left\{x\in V|\phi \right\}=\left\{X\right\}\leftrightarrow \forall x\left(\left(x\in V\wedge \phi \right)\leftrightarrow x=X\right)$$