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I just want to double check that this is true. Google doesn't cooperate well with equations.

P(A,B|C) = P(A|B,C)P(B|C)

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Indeed, because

$$ \frac{P(A,B,C)}{P(C)} = \frac{P(A,B,C)}{P(B,C)}\frac{P(B,C)}{P(C)} $$

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