Since $X$ and $Y$ are independent, then $\mathbb{P}(X \in E, Y \in F) = \mathbb{P}(X \in E) \mathbb{P}(Y \in F)$ for every borel set $E, F \subseteq \mathbb{R}$. Thus, if $E, F \subseteq \mathbb{R}$ are borel sets, then so are $\frac{E - \beta}{\alpha}$ and $\frac{F - \delta}{\gamma}$, whence \begin{equation} \begin{split} \mathbb{P}(\alpha X + \beta \in E, \gamma Y + \delta \in E) & = \mathbb{P} \left( X \in \frac{E - \beta}{\alpha}, Y \in \frac{F - \delta}{\gamma} \right) \\ & = \mathbb{P} \left( X \in \frac{E - \beta}{\alpha} \right) \mathbb{P} \left(Y \in \frac{F - \delta}{\gamma} \right) \\ & = \mathbb{P}(\alpha X + \beta \in E) \mathbb{P}(\gamma Y + \delta \in E) \end{split} \end{equation} $\square$