Efficient unbiased estimator implies exponential family

In a regular one-parameter statistical model, equality in the Cramér-Rao lower bound for an unbiased estimator $T$ implies the score function ${\partial }_{\theta }\log f(x;\theta )=I(\theta )(T(x)-\theta )$. Integrating in the parameter on a regular connected interval gives $f(x;\theta )=h(x)\exp (A(\theta )T(x)-B(\theta ))$, with $A^{\prime }=I$ and $B^{\prime }=\theta I$. Thus efficiency for every parameter value forces an exponential family structure, under the usual common-support and differentiation assumptions.