I believe the following is a theorem of Skolem. How is it proved, or where may I find a proof?
Suppose $\mathscr A$ is a formula of first-order logic in which no constants or function symbols occur but in which there are $n$ distinct predicate symbols $P_k(x_1,\dots,x_{a(k)})$, $k=1,\dots,n$. ($P_k$ is $a(k)$-ary, and each $P_k$ may occur more than once in $\mathscr A$.) Then there is a formula $\mathscr B$ of first-order logic which contains only unary and binary predicate symbols such that $\vDash\mathscr A$ iff $\vDash\mathscr B$.
The claim is demonstrated by replacing each occurrence of $P_k(x_1,\dots,x_{a(k)})$ in $\mathscr A$ with $$(\forall x)((B_1(x_1,x)\wedge B_2(x_2,x)\wedge\dots\wedge B_{a(k)}(x_{a(k)},x))\to U_k(x))$$ to make $\mathscr B$, where $B_1,\dots,B_{a(k)}$ are new binary predicate symbols and $U_k$ is a new unary predicate symbol.