Modalized fixed-point theorem for GL ($GL⊢F\leftrightarrow X(F)$)

If every occurrence of $p$ in $X(p)$ lies under a box, Gödel-Löb logic has a formula $F$ without $p$, using only the other letters of $X$, satisfying the displayed equivalence. Write $X(p)=H(\square C_1(p),\dots ,\square C_m(p))$ using the outermost boxes that contain $p$, so $H$ has no $p$. Solve $q_i\leftrightarrow \square C_i(H(q_1,\dots ,q_m))$ by simultaneous boxed fixed-point elimination in GL. If the solutions are $Q_i$, then $F=H(Q_1,\dots ,Q_m)$ works by replacement of provably equivalent formulas.