Converting If-else integer equation to Linear Programming
Your condition is effectively excluding $(0,1,1)$ and $(1,0,1)$ corners of the unit cube. Thinking of it pictorially gives you a quick formulation: (i) construct the cube, (ii) chop those corners off and (iii) ensure integrality.
In (ii), the corners we want to get rid of are on the $BC$ and $AC$ planes. We can cut from the correctly aligned diagonals on those planes. However, moving in $+A$ direction, the cut should shrink and vanish at $A=1$ to include $(1,1,1)$.
\begin{align*}
&amp;\text{(i) } 0 \leq A,B,C \leq 1  \\
&amp;\text{(ii) }-A+B+C \leq 1, A-B+C \leq 1 \\
&amp;\text{(iii) }A,B,C \in \{0,1\}
\end{align*}