Trying to prove this to myself I note that the density of the normal distribution is fully specified ones variance and mean is specified and does not depend on x hence I can reason:

\(f(e) = \int f(e,x) dx = \int f(e\lvert x) f(x) dx = f(e\lvert x) \int f(x) dx = f(e\lvert x)\)

is that the way to go?