The exercise is given as it follows:
If $G$ is connected with order at least $3$, then $G$ is a block iff for all edge $e \in E(G)$ and for each $\{u,v\} \in V(G)$ exists a walk $W$ such that it does not contain $e$.
Prooving that if $G$ is a block $\implies [...]$ it's not so difficult following that if $G$ is a block then exist a cycle such that $\{u,v\}$ is contained within. But I'm stuck prooving that for all edge [...] $\implies G$ is a block. Any hints?