Minimizing the full adder - where did this XOR come from?
Consider the truth table for $(\overline{B}\overline{C}+BC)$. It has 1's exactly when $B$ and $C$ get the same value, which is exactly when $(B\oplus C)$ gets 0, hence, $\overline{B\oplus C}$ is equivalent to $(\overline{B}\overline{C}+BC)$.