Implicit neural SLAM has achieved remarkable progress recently. Nevertheless, existing methods face significant challenges in non-ideal scenarios, such as motion blur or lighting variation, which often leads to issues like convergence failures, localization drifts, and distorted mapping. To address these challenges, we propose $\textbf{EN-SLAM}$, the first event-RGBD implicit neural SLAM framework, which effectively leverages the high rate and high dynamic range advantages of event data for tracking and mapping. Specifically, EN-SLAM proposes a differentiable CRF (Camera Response Function) rendering technique to generate distinct RGB and event camera data via a shared radiance field, which is optimized by learning a unified implicit representation with the captured event and RGBD supervision. Moreover, based on the temporal difference property of events, we propose a temporal aggregating optimization strategy for the event joint tracking and global bundle adjustment, capitalizing on the consecutive difference constraints of events, significantly enhancing tracking accuracy and robustness. Finally, we construct the simulated dataset $\textbf{DEV-Indoors}$ and real captured dataset $\textbf{DEV-Reals}$ containing 6 scenes, 17 sequences with practical motion blur and lighting changes for evaluations. Experimental results show that our method outperforms the SOTA methods in both tracking ATE and mapping ACC with a real-time $17$ FPS in various challenging environments. The code and dataset will be released soon.