I know and have proved them following theorem:
Let $G$ be a 2-connected graph. Then for any 2 distinct vertices $x$ and $y$ , there exist $2$ Internally disjoint $(x-y)$-paths.
I changed the statement a little:
‘Let $G$ be a connected graph. Suppose $G$ has no bridge. Show that for any 2 distinct vertices $x$ and $y$ , there exist $2$ edge-disjoint $(x-y)$-paths. ’
I want to prove or disprove this but was not able to do it. It seems correct(I created many examples that agrees with it) but I am not certain.
I tried mathematical induction(on the distance between $x$ and $y$) and method by contradiction.
what you are referring to is a special case of Manger's theorem. https://en.wikipedia.org/wiki/Menger%27s_theorem