Self-quoting fixed point

Let $A$ code application and $C$ code numeral quotation. For a lambda term $F$, choose a fresh $x$ and put $H=\lambda x.F(Ax(Cx))$ and $Q=Hd(H)$. Then $Ad(H)(Cd(H)){\equiv }_{\beta }d(Q)$, so $Q{\equiv }_{\beta }Fd(Q)$. This diagonal argument uses the syntax of the actual term $Q$, including any free variables, not an equivalence-class code.