Consider $Y_1,..., Y_n$ i.i.d. standard normal. Now for some $m>0$ I want to know about
$P(Y_1 \leq t \mid \overline{Y_n} \geq m)$
where $\overline{Y_n}$ is the empirical mean. Intuitivly this should be (or at least converge to as $n \to \infty$) $P(Z \leq t)$ with $Z$ a normal with mean $m$ and variance $1$.
Is this correct and if so, how to prove it?