Below you can find some references. They do not give the result directly in the form you want but instead give more general results where $E$ is assumed to be a Pták space (= B-complete = fully complete) which then gives the result you want by the implication Fréchet $\Rightarrow$ Pták.
[1] J. Horváth: Topological Vector Spaces and Distributions, Theorem 3.17.2, p. 296.
[2] H. Jarchow: Locally Convex Spaces, Theorem 9.7.1, p. 186.
[3] H.H. Schaefer: Topological Vector Spaces, Theorem 8.3, p. 163.