Simultaneous boxed fixed-point elimination in GL ($F_i\leftrightarrow \square C_i(F_1,\dots ,F_m)$)

In Gödel-Löb logic, the one-variable identity $\square C(\top )\leftrightarrow \square C(\square C(\top ))$ solves a boxed equation with a formula omitting its unknown letter. A finite simultaneous system can be solved by elimination. First solve its last equation with the other unknowns treated as parameters, substitute that formula into the preceding equations, and apply induction. Substituting the resulting solutions back into the last formula completes the system. This construction introduces no new parameter letters.

Let $U=\square Y(\top )$ and $V=\square Y(U)$. Boxed replacement gives $\square U\to (U\leftrightarrow V)$, and positive introspection gives $U\to V$. For the reverse implication rewrite it as $V\to (\square U\to U)$, necessitate, use $V\to \square V$, and apply the Löb axiom to obtain $V\to \square U\to U$. This explicit one-variable construction needs no reflection axiom.