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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.

The language of Gödel-Löb logic has propositional variables, $\perp$, implication, and the unary operator $\square$; the other logical connectives are abbreviations. Its axioms are all classical tautologies, the distribution scheme $\square (A\to B)\to (\square A\to \square B)$, and the Löb scheme $\square (\square A\to A)\to \square A$. Its rules are modus ponens rule, necessitation rule of theorems, and uniform substitution.

For a single boxed equation $p\leftrightarrow \square C(p)$, set $F=\square C(\top )$. The identity supplied in the source gives directly $GL⊢F\leftrightarrow \square C(F)$. If $p$ does not occur, take $F=\square C$ instead.

Proceed by induction for a system of $m$ equations, allowing other propositional variables as parameters. Treat $p_1,\dots ,p_{m-1}$ as parameters in the last equation and putThe one-variable result proves $D\leftrightarrow \square C_m(p_1,\dots ,p_{m-1},D)$. Substitute this $D$ for $p_m$ in the first $m-1$ equations. They are still boxed equations, so induction supplies solutions $F_1,\dots ,F_{m-1}$. Define $F_m=D(F_1,\dots ,F_{m-1})$.

By construction, the first $m-1$ equations hold after this substitution; substituting into the theorem for $D$ proves the last. HenceThe elimination introduces no new parameter letters and removes every unknown letter. This gives a constructive simultaneous boxed fixed-point elimination in GL.