$X_1,X_2,\ldots,X_n$ are independently and identically $N(\mu,\sigma^2)$ distributed. I want to find the UMVUE of $P(X \le c)$.
$T=({\bar X},S^2)$ is sufficient and complete for $(\mu,\sigma^2)$. It is easy to find an unbiased estimator $I(X_1 \le c)$, so the answer is $E(I(X_1 \le c) \mid T)$.
$A=\frac{X_1-{\bar X}}{S}$ is an ancillary statistic of $(\mu,\sigma^2)$, so it is independent of $T$.
$$\begin{align} E(I(X_1 \le c) \mid T) & = P(X_1 \le c \mid T) \\ & = P(A \le \frac{c-{\bar X}}{S} \mid T) \\ & = P(A \le \frac{c-{\bar x}}{s}) \\ \end{align}$$
The question is what is the pdf of $A$.