\begin{align*} P(A) & + P(B) - P(A \cap B) \\ & = [P(A-B) + P(A \cap B)] \\ & + [P(A \cap B) + P(B-A)] - P(A \cap B)\\ & = P(A-B) + P(A \cap B) + P(B-A)\\ & = P(A \cup B) \end{align*}