Prove the identity: $\neg((a\wedge b)\vee(a\wedge\neg c)) = (\neg a\wedge b)\vee(\neg b\wedge\neg c)$

Question

I've been trying to prove this, but I have no clue how to take it to the end. I can go up to the part:

$$\neg a\wedge\neg b\wedge (c\vee\neg c)\vee\neg a\wedge c\vee\neg b \wedge\neg c$$

After this, I do not understand how to get rid of $\neg a\wedge\neg b$.

I would highly appreciate anyone who could explain this. Thank you in advance :)

Answer 1

As @tia noted, if $\neg a\land\neg b\land c$ the left-hand side is true while the right-hand side is false. The left-hand side is$$\begin{align}\neg(a\land b)\land\neg(a\land\neg c)&\equiv(\neg a\lor\neg b)\land (\neg a\lor c)\\&\equiv\neg a\lor(\neg b\land c).\end{align}$$The right-hand side is already as simple as it can get.