Assuming that the functional integral of a functional derivative is zero, so
$$ \int \mathcal{D}[\phi] \frac{i}{\hbar}\left\{ \frac{\delta S[\phi]}{\delta \phi}+J(x) \right\}\exp \left[ {i \over \hbar} \left( S[\phi]+\int \! \mathrm{d}^4 x \, J(x)\phi(x)\right)\right]=0. $$
Apart from this trivial equation what else do I need to prove the Dyson-Schwinger equations?
To formally show the Schwinger-Dyson equations, use the fact the
$$\begin{align}0~=~&\int [d\phi]\frac{\delta}{\delta \phi^{\alpha}(x)} \left\{F[\phi]\exp\left[\frac{i}{\hbar}\left(S[\phi]+\int \!d^nx^{\prime}~J_{\alpha}(x^{\prime})\phi^{\alpha}(x^{\prime}) \right)\right]\right\}\cr ~=~&\int [d\phi]\left\{\frac{\delta F[\phi]}{\delta \phi^{\alpha}(x)} + \frac{i}{\hbar}F[\phi]\left(\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}+J(x)\right)\right\}\cr &\qquad \exp\left[\frac{i}{\hbar}\left(S[\phi]+\int \!d^nx^{\prime}~J_{\alpha}(x^{\prime})\phi^{\alpha}(x^{\prime}) \right)\right]\cr ~\propto~&\left<\frac{\delta F[\phi]}{\delta \phi^{\alpha}(x)} + \frac{i}{\hbar}F[\phi]\left(\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}+J(x)\right)\right>_{\!J} , \end{align}$$
cf. this Phys.SE post. Without specifying the action $S[\phi]$ and field content $\phi^{\alpha}$ further, it is impossible to provide a rigorous proof, since the path integral is not properly defined in the first place. See also this related Phys.SE post.