I have this expression and have to simplify it : $\lnot B\,\lnot C + A \,\lnot B \, C$
What I did was: $\lnot B\,\lnot C + A \,\lnot B \, C = \ \lnot B(\lnot C + C + A) = \lnot B(1) = \lnot B$
Why isn't this simplification correct? First I applied the commutative property so that I could use the distributive as I had a $\lnot B$ on the both sides; then by complements, $\lnot C + C = 1$, and by identity $\lnot B(1) = \lnot B$. Supposedly, it's not right; what is the error?
The error lies in the application of the distributive property: $$\neg B\neg C + A \neg B C = \neg B (\neg C + A\cdot C).$$ Notice how your $+$ became a $\cdot$ in my version.