Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 76 / 5 / ii / Solution

Choose $x$ fresh for $F$ and put $H=\lambda x.F(Ax(Cx))$, $Q=Hd(H)$. The preceding coding equations giveConsequentlyThis self-quoting fixed point is valid even if $F$ has free variables, since the effective Gödel numbering codes open lambda terms too. Freshness of $x$ prevents variable capture.