If I have a conditional probability matrix for binary variables $A,B,C$ with entries of the form $$ P(A | B \cap C) = \left( \begin{matrix} P(a_1 | b_1 \cap c_1) & P(a_1| b_1 \cap c_2) & P(a_1|b_2 \cap c_1) & P(a_1 | b_2 \cap c_2) \\ P(a_2 | b_1 \cap c_1) & P(a_2| b_1 \cap c_2) & P(a_2|b_2 \cap c_1) & P(a_2 | b_2 \cap c_2) \end{matrix} \right) $$
how can I use this matrix to obtain the matrix $$ P(A | C) = \left( \begin{matrix} P(a_1 | c_1) & P(a_1| c_2) \\ P(a_2 | c_1) & P(a_2| c_2) \end{matrix} \right) $$
where $A$ is assumed to be independent of $B$?
Clearly in general $P(X | Y \cap Z) = P(X|Z)$ under the assumption that $X$ and $Y$ are independent, but I am confused as to how the entries of the matrices above are related.