Minimizing the full adder - where did this XOR come from?

When minimizing the full adder, I don't understand why $A(\bar{B}\bar{C} + BC)$ reduces to $A\overline{(B\oplus{C})}.$

$(\bar{B}\bar{C} + BC)\to (B\oplus{C})$ is partially decipherable, but why is $(B\oplus{C})$ inverted to $\overline{(B\oplus{C})}?$

Full adder simplification:

$
\bar{A}\bar{B}C + \bar{A}B\bar{C} + A\bar{B}\bar{C} + ABC \\
= \bar{A}(\bar{B}C + B\bar{C}) + A(\bar{B}\bar{C} + BC) \\
= \bar{A}(B\oplus{C}) + A(\overline{B\oplus{C}}) \\
= A\oplus{(B\oplus{C})}
$

Could you help me out?

PS: I hope that this is the correct subforum of StackExchange to ask this (perhaps Electrical Engineering is the proper venue). I couldn't find appropriate tags on either site.

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 $(\overline{B}\overline{C}+BC)$.

StackExchange Computer Science Q#9866, answer score: 5