An identity involving expectations of products of the normal CDF

Question

Let $\Phi$ be the standard normal CDF and let $X$ be standard normal. Suppose $\sigma_1,\sigma_2,\sigma_3$ are positive constants. Show that $$ E \Bigl[\Phi\Bigl(\frac{\sigma_1}{\sigma_2}X\Bigr) \Phi\Bigl(\frac{\sigma_1}{\sigma_3}X\Bigr)\Bigr] + E \Bigl[\Phi\Bigl(\frac{\sigma_2}{\sigma_1}X\Bigr) \Phi\Bigl(\frac{\sigma_2}{\sigma_3}X\Bigr)\Bigr] + E \Bigl[\Phi\Bigl(\frac{\sigma_3}{\sigma_2}X\Bigr) \Phi\Bigl(\frac{\sigma_3}{\sigma_1}X\Bigr)\Bigr] = 1. $$ Numerical calculations show that this should be true for arbitrary constants. Each expectation can be obviously expressed as probabilities involving three standard normals but this does not seem to help. I assume I'm missing something obvious.