Definition:Restricted Universal Quantifier

Definition
Let $A$ be a class in ZF.

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


 * $\forall x \in A: P \left({x}\right) \quad \text{for} \quad \forall x: \left({x \in A \implies P \left({x}\right)}\right)$

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