Definition:Restricted Existential Quantifier

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Let $A$ be a class in ZF.

The restricted existential quantifier is denoted $\exists x \in A$ and is defined as the following definitional abbreviation:

$\exists x \in A: P \left({x}\right) \quad \text{for} \quad \exists x: \left({x \in A \land P \left({x}\right)}\right)$

where $P \left({x}\right)$ is any well-formed formula of the language of set theory.