Axiom:Axiom of Replacement/Set Theory

Let $$P \left({y, z}\right)$$ be a propositional function, which determines a function.

For any set $$S$$, there exists a set $$x$$ such that, for any element $$y$$ of $$S$$, if there exists an element $$z$$ satisfying the condition $$P \left({y, z}\right)$$, then such $$z$$ appear in $$x$$.

$$\exists x: \forall y \in S: \left({\exists z: P \left({y, z}\right) \Longrightarrow \exists z \in x: P \left({y, z}\right)}\right)$$