Independence of variables in Convolution

[redtree]

Given a convolution:
\begin{equation}
\begin{split}
g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz
\end{split}
\end{equation}

Do $z$ and $x$ have to be independent? For example, can one set $x=z+y$ such that:
\begin{equation}
\begin{split}
\int_{-\infty}^{\infty} g(z) h(x-z) dz&=\int_{-\infty}^{\infty} g(z) h(y) dz
\end{split}
\end{equation}

 

[FactChecker]

The variable z is a temporary variable for the integral. You can not make x depend on it.