= Solution
Work in the ambient universe and let $a,u\in L$. Suppose
$$
L\models\forall x\in a\;\exists!y\;\varphi(x,y,u).
$$
For each $x\in a$, let $y_x$ be this unique witness. The relativization $\varphi^L$ is a first-order formula, so the ambient <Axiom schema of replacement> collects the witnesses $y_x$ into a set. Every witness lies in the <constructible hierarchy>, hence there is an ordinal $\alpha$ such that
$$
\{y_x:x\in a\}\subseteq L_\alpha.
$$
For example, take the supremum of one constructible rank for each witness and then increase it by one.
The set $L_\alpha$ itself belongs to $L_{\alpha+1}\subseteq L$. Taking $b=L_\alpha$, every $x\in a$ has a witness $y\in b$ satisfying $\varphi^L(x,y,u)$. Therefore
$$
L\models\exists b\;\forall x\in a\;\exists y\in b\;\varphi(x,y,u),
$$
which is the stated instance of Replacement.
Back to article page
