Separation in the constructible universe

Choose an ordinal bound large enough that one level contains $a$ and every $p_i$. Reflect $\varphi$ above this bound, obtaining a nonempty transitive $L_{\beta }$ containing those parameters. For every $x\in a$, transitivity puts $x$ in $L_{\beta }$, so ${\varphi }^L(x,p)$ agrees with satisfaction of $\varphi (x,p)$ in $L_{\beta }$.

Over $L_{\beta }$ the formula $x\in a\wedge \varphi (x,p)$ defines exactly the desired subset: the first conjunct uses actual membership and the second agrees by step 1.1. F2 puts this subset in $\mathrm{Def}(L_{\beta })=L_{\beta +1}\subseteq L$. An empty a or a formula with no satisfying elements gives the empty subset by the same definition. The argument uses only this fixed formula and ambient ZF.