Simplifying a Boolean Expression with two additional conditions
This is a super long detour for a simple result:


A·B = 0 implies that at most one of A, B is true
A+B = 1 implies that at least one of A, B is true
This implies that exactly one of A, B is true.
This implies that \$\overline A = B\$.


Then

\begin{align}
(A + C) \cdot( \overline A + B ) \cdot (B + C) 
&amp;=(\overline B +C)\cdot(B+B)\cdot (B+C)\\
&amp;=B(\overline B +C)(B+C)\\
&amp;=(B\overline B + BC)(B+C)\\
&amp;=(0+BC)(B+C)\\
&amp;= BCB + BCC\\
&amp;=BC\\
&amp;\overset!=BC 
\end{align}