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.

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.