Let $G$ be a $k$-regular bipartite graph with $k\ge3$, and $e_{1},e_{2}$ be edges of $G$. Show that if $G-\{e_{1},e_{2}\}$ is connected, then there exist a perfect matching in $G$ that includes $e_{1}$ but not $e_{2}$.
I have been able to demonstrate that there exists a perfect matching that includes $e_{1}$ and that there exists another perfect matching that does not includes $e_{2}$ using Hall's theorem. However, I cannot find a single matching that satisfies both conditions at the same time.
I appreciate any hints.