Suppose $X$ and $Y$ are independent normal random variables with means $\mu_X, \mu_Y$ and standard deviations $\sigma_X, \sigma_X$. I want to determine if there is a closed-form expression for $P(X \leq x \mid X < Y)$, where $x$ is a constant.
I know that the distribution $X - Y$ is normally distributed with mean $\sigma_X - \sigma_Y$ and standard deviation $\sqrt{\sigma_X^2 + \sigma_Y^2}$, so the question boils down to understand the quantity $P(X \leq x \cap X < Y)$.
I've tried looking at these quantities using the law of total probability, conditioning over $Y=y$ (as per this question), but haven't had any luck obtaining a closed-form expression. We can get a general expression for arbitrary random variables, but this involves a fairly nasty integral that I don't know how to approach (namely, the product of the CDF and the PDF of two different distributions): $$\int_{-\infty}^x \text{CDF}_X(y) \cdot \text{PDF}_Y(y) \,dy$$ This question studies this integral for a standard normal distribution, where the integral is over the whole real line, but the integral I'm concerned with ranges from $-\infty$ to $x$ and involves non-standard normal distributions. Wikipedia has a list of integrals that look similar to the above expression, but none that match what I need. I've skimmed the referenced paper on that page, but no joy there either.
Whilst I feel finding a closed-form expression for the above integral would certainly work, I wonder if I may be missing a slicker, more intuitive way of answering this question.
Any insights or pointers in the right direction would be very much appreciated! Many thanks in advance.