Diagonal lemma with parameters

Statement

Let $\psi (x, y_1, \dots , y_n)$ be any formula with free variables $x, y_1, \dots , y_n$.

Then there is a formula $\varphi (y_1, \dots y_n)$ with free variables $y_1, \dots , y_n$ such that $T \vdash \varphi (y_1 , \dots , y_n) \leftrightarrow \psi (\ulcorner \varphi (y_1 , \dots , y_n) \urcorner, y_1 , \dots , y_n)$.