I'm re-reading a paper of Stevo Todorcevic's entitled "Localized Reflection and Fragments of PFA" and there's a claim in the proof of one of the lemmas that I thought I understood but now I'm not so sure. The claim is this:

Suppose $0^{\sharp}$ does not exist, and let $a \in L_{\omega_2}$, $\varphi$ a formula, $\theta$ a regular cardinal in $L$ such that $L_{\theta} \vDash \varphi (a)$. Then there exists $\lambda$, a cardinal in $V$, such that $\lambda ^+ = (\lambda ^+)^L$ and $L_{\lambda ^+} \vDash \varphi (a)$.

Why is this true?

I'm re-reading a paper of Stevo Todorcevic's entitled "Localized Reflection and Fragments of PFA" and there's a claim in the proof of one of the lemmas that I thought I understood but now I'm not so sure. The claim is this:

Suppose $0^{\sharp}$ does not exist, and let $a \in L_{\omega_2}$, $\varphi$ a formula, $\theta$ a regular cardinal in $L$ such that $L_{\theta} \vDash \varphi (a)$. Then there exists $\lambda$, a cardinal in $V$, such that $\theta \leq \lambda ^+ = (\lambda ^+)^L$ and $L_{\lambda ^+} \vDash \varphi (a)$.

Why is this true?