I realize I'm a couple weeks behind and nobody might read this, but this is false (e.g., if I flip two fair + independent coins and A = "first coin is heads" and B = "second coin is heads", P(B) = 1/2, but P(B|A) x P(A) = (1/2)x(1/2) = 1/4). I think what you mean is P(A and B) = P(B|A) x P(A).
(unless I'm missing some context and we are assuming B is a subset of A, but if so I'm not sure why we are assuming that)
Direct link
Want more? Brainstorm more spells in the comments and I'll (likely) do more of these!