A question on non-standard ordinals in $\alpha-$recursion

Let $M$ be an admissible set, namely, $M\models KP$ where KP stands for axioms of Kripke–Platek set theory. Denote $\beta=M\cap ORD$ where $ORD$ is the class of ordinals. I wanted to prove $L_\beta\models KP$ where $L_\beta$ is the Godel Constructible universe at level $\beta$. The difficulty I encountered was in the verification of $\Delta_0$ replacement. Suppose $L_\beta \models \forall u \forall x\in u \exists y \varphi(x,y)$ I want to show for any $u\in L_\beta$ $L_\beta\models \exists v \forall x\in u \exists y\in v \varphi(x,y)$. We could apply $\Delta_0$ comprehension in $M$, however, we may get a $b\in M\backslash L_\beta$. I sort of know it has something to do with the nonstandard ordinals (ordinals not in $L_\beta$) in $M$, but I don't know how to get the arguments. Please help! Thanks in advance.

• This is lemma II 7.1 in K. Devlin "Constructibility". – Eran Apr 13 '13 at 16:58
• @Eran: Thanks a lot! – Jing Zhang Apr 13 '13 at 19:02