In this paper, we aim at analyzing the classical information spreading \push protocol in \emph{dynamic} networks. We consider the \emph{edge-Markovian} evolving graph model which captures natural temporal dependencies between the structure of the network at time $t$, and the one at time $t+1$. Precisely, a non-edge appears with probability $p$, while an existing edge dies with probability $q$. In order to fit with real-world traces, we mostly concentrate our study on the case where $p=\Omega(\frac{1}{n})$ and $q$ is constant. We prove that, in this realistic scenario, the \push protocol does perform well, completing information spreading in $O(\log n)$ time steps, w.h.p., even when the network is, w.h.p., disconnected at every time step (e.g., when $p\ll \frac{\log n}{n}$). The bound is tight. We also address other ranges of parameters $p$ and $q$ (e.g., $p+q=1$ with arbitrary $p$ and $q$, and $p=\Theta\left(\frac{1}{n}\right)$ with arbitrary $q$). Although they do not precisely fit with the measures performed on real-world traces, they can be of independent interest for other settings. The results in these cases confirm the positive impact of dynamism.