The law of total probability applied to a conditional probability

Dear future self,

I’ve just lost (again) about half an hour of my life trying to find a vaguely remembered formula that generalizes the law of total probability to the case of conditional probabilities. Here it is. You’re welcome.

So what is the probability of dying from a lighting strike if you’re an American who knows this statistic?

The law of total probability says that if you can decompose the set of possible events into disjoint subsets (say $B$ and $\overline{B}$), then (with obvious generalization to more than two subsets):

$$\Pr(A) = \Pr(A \mid B) \Pr(B) + \Pr(A \mid \overline{B}) \Pr(\overline{B})$$

But what if you’re dealing with $\Pr(A \mid C)$ instead of just $\Pr(A)$? What’s the formula for the law of total probability in that case? What you’re searching for can be found by googling for “total law probability conditional”:

$$\Pr(A \mid C) = \Pr(A \mid B, C) \Pr(B \mid C) + \Pr(A \mid \overline{B}, C) \Pr(\overline{B} \mid C) $$

There’s a great derivation here: https://math.stackexchange.com/questions/2377816/applying-law-of-total-probability-to-conditional-probability.