Simultaneous diagonal lemma

Statement

Let $\psi_1 (x_1 , x_2)$ and $\psi_2 (x_1 , x_2)$ be formulae with free variables $x_1$ and $x_2$.

Then there are sentence $\varphi_1$ and $\varphi_2$ such that $T \vdash \varphi_1 \leftrightarrow \psi_1(\ulcorner \varphi_1 \urcorner, \ulcorner \varphi_2 \urcorner)$ and $T \vdash \varphi_2 \leftrightarrow \psi_2(\ulcorner \varphi_1 \urcorner, \ulcorner \varphi_2 \urcorner)$.

The case with $n$ many formulae is similar.