Statistical Inference — Rao-Blackwellisation properties

Let $U = U(X_1, X_2, X_3, …, X_n)$ be an unbiased estimator of $\gamma(\theta)$ and $T$ be a sufficient statistic for $\theta$. Define $\phi(t) = E[U \mid T = t]$. Consider the following statements: I. $\phi(t)$ is always independent of $\theta$. II. The variance of $\phi(t)$ is less than the variance of $U$. Which of the statements given above is/are correct?