---
tag: P3XFQC6ACX
kind: statement
role: lemma
name: Diagonal lemma with parameters
sources:
  - wikipedia: en:348969@r1344931050
    locator: Diagonal lemma, section Some generalizations > Diagonal lemma with parameters
    note: adopted
    mention: wikipedia://en/348969#CABK
    sections: [Statement]
inventory: [RBPXACPVEP]
---

## 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 $T \vdash \varphi (y_1 , \dots , y_n) \leftrightarrow \psi (\ulcorner \varphi (y_1 , \dots , y_n) \urcorner, y_1 , \dots , y_n)$.