Let $Z \in \text{Gaussian}(0, 1)$ be a
D211: Standard gaussian random real number. Now $\sigma Z \in \text{Gaussian}(0, \sigma^2)$ and $\sigma Z + \mu \in \text{Gaussian}(\mu, \sigma^2)$. Result
R3680: Characteristic function of standard gaussian random real number shows that
\begin{equation}
\mathbb{E} (e^{i t Z})
= e^{- t^2 / 2}
\end{equation}
Thus
\begin{equation}
\mathfrak{F}_{\sigma Z}(t)
= \mathbb{E}(e^{i t a Z})
= \mathfrak{F}_Z(\sigma t)
= e^{- (\sigma t)^2 / 2}
= e^{- \sigma^2 t^2 / 2}
\end{equation}
Next, result
R4639: Characteristic function for translated random real number shows that
\begin{equation}
\mathfrak{F}_{\sigma Z + \mu} (t)
= e^{i t \mu} \mathfrak{F}_{\sigma Z} (t)
= e^{i t \mu} e^{- \sigma^2 t^2 / 2}
= e^{i t \mu - \sigma^2 t^2 / 2}
\end{equation}
To establish the result, it is only left to confirm that
\begin{equation}
\mathfrak{F}_{\sigma Z + \mu} (t)
= \mathfrak{F}_X (t)
\end{equation}
for every real number $t \in \mathbb{R}$. Since $\sigma Z + \mu \overset{d}{=} X$, this is guaranteed by
R2405: Characteristic function uniquely identifies the distribution of a random real number. $\square$
