Take two random variables $X=a+bX_0$ and $Y=c+dY_0$, and define $T=X-Y=\mu+\sigma Z$ where $\mu$ is the mean of $T$, $\sigma$ its standard deviation and $Z$ is a standardized random variable with mean $zero$, unit variance and continuous PDF $\phi$ and continuous and increasing CDF $\Phi$. NB: $a,b,c,d \in \mathbb{R}$
What does $\Phi(\frac{-\mu}{\sigma})$ mean, either in terms of $T$? or $X$ and $Y$?
It represents the probability that $T$ is less than $0$ or equivalently the probability that $X$ is less than $Y$.
$$\text{Prob}(T \le 0) = \text{Prob}(\mu+\sigma Z \le 0)=\text{Prob}(Z \le \frac{-\mu}{\sigma})=\Phi(\frac{-\mu}{\sigma})$$