I am reading the lecture note from Cambridge University about Probabilistic Ranking and they claim that the normalized constant has a closed form in the below formula but I could not know how to prove. Would you please help me to handle this relation?

$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \mathcal{N}(w_1; \mu_1, \sigma^2_1)\mathcal{N}(w_2; \mu_2, \sigma^2_2) \Phi(y(w_1 - w_2)) \, \mathrm{d}w_1 \mathrm{d}w_2 = \Phi ( \frac{y(\mu_1 - \mu_2)}{\sqrt{1+ \sigma^2_1 + \sigma^2_2}})$

In this case, $\Phi(t) = \int_{-\infty}^t \mathcal{N}(x; 0, 1)\, \mathrm{d}x$ and $\mathcal{N}$ represents the normal distribution. $y$ is the observation and its value is +1 or -1.

One hint that they give is

$\int_{-\infty}^{\infty} \delta(y - sign(t)) \mathcal{N}(t; \mu, \sigma^2) \,\mathrm{d}t = \Phi(\frac{y\mu}{\sigma}) $

$ \Phi(y(w_1 - w_2)) = \int_{-\infty}^{\infty} \delta(y-sign(t)) \mathcal{N}(t; w_1 - w_2,, 1) \, \mathrm{d}t$.

$\delta$ is Dirac delta function which is use as indicator in the case of continuous variables.

However, I get stuck and dont know how to prove this. Would you please help me to prove?

Thank you.


P/S: The link of the slide is http://mlg.eng.cam.ac.uk/teaching/4f13/1213/lect0607.pdf and this course is over already.