I have an Integral:
$$ \int_{-\infty}^{-y_1} \Phi(y_2)d\Phi(x_1) $$
Here:
$\Phi(y_2)$ is the Gaussian density function of variable '$y_2$' which has to be integrated w.r.t Gaussian density of another '$x_1$' and the upper limit is another variable $y_1$.
I have the identity: $$y_2 = y_1 + x_1$$
Essentially $x_1$, $y_1$, are arguments of the respective PDFs of two independent Normal Random Variables $X_1$ and $Y_1$ and integral becomes:
$$ \int_{-\infty}^{-y_1} \Phi(y_1+x_1)d\Phi(x_1) $$
Now, changing $\Phi(y_1+x_1) = u$, I have: $$ d\Phi(x_1) = du\tag{A}\\ $$ When $x_1=-\infty,\; \Phi(-\infty)=0=u\tag{B}$ $x_1=-y_1,\; \Phi(0) = \frac{1}{2}=u\tag{C}$
Integral becomes: $$ \int_0^{1\over2}udu $$
Which seems to work for my calculations but, essentially, are steps (A) through (C) correct?