Let $P_{\theta}$ be the distribution (known up to a parameter $\theta$ in parameter space $\Theta$) of a random variable.
A parameter (function) $\gamma=g\left(\theta\right)$ is called identifiable if and only if $$\forall\,\theta,\theta^{*}\in\Theta:P_{\theta}=P_{\theta^{*}}\Rightarrow g\left(\theta\right)=g\left(\theta^{*}\right).$$
A parameter (function) $\gamma=g\left(\theta\right)$ is called estimable if and only if $$\exists\,\hat{\gamma}:\mathbb{E}_{\theta}\left[\hat{\gamma}\right]=g\left(\theta\right)\:\forall\,\theta\in\Theta,$$ where $\hat{\gamma}$ is an unbiased estimator of $\gamma$.
Why does estimability imply identifiability?