Confused with probabilistic notation and precedence

Question

Is $P(A|B,C) = P(A|(B \cap C))$ or $P((A|B) \cap C)$?

In my book they do something like this in a proof regarding Markov models:

$P(A|B_{1:t+1}) = P(A|B_{1:t},B_{t+1}) = P(B_{t+1}|A,B_{1:t})*P(A|B_{1:t})$/$P(B_{1:t+1})$

And I'm totally confused. Could someone write this out with parenthesis and $\cap/\cup$?

edit: Removed the $\alpha$

Answer 1

Whenever $P(B\cap C)>0$ you have $$ P(A\mid B,C)=P(A\mid B\cap C)=P(A\cap B\cap C)/P(B\cap C). $$ The "quantity" $P((A\mid B)\cap C)$ is nonsense, since $A\mid B$ isn't a set.