Law of Excluded Middle in Logic Proof

Question

I'm having some difficulty doing a proof for the following:

$$\neg A \vee \neg(\neg B \wedge (\neg A \vee B))$$

It is said that you could use the law of excluded middles.

Any help or guidance would be appreciated. Thanks in advance!

Answer 1

Consider this: $$ \begin{align*} \neg A\lor\neg(\neg B\land(\neg A\lor B)) &\equiv \neg A\lor(B\lor\neg(\neg A\lor B))\\ &\equiv \neg A\lor(B\lor (A\land\neg B)) \\ &\equiv \neg A\lor((B\lor A)\land(B\lor\neg B)) \\ &\equiv \neg A\lor((B\lor A)\land \top) \\ &\equiv \neg A\lor(B\lor A) \end{align*} $$ where $B\lor\neg B\equiv\top$ by the law of excluded middle. Applying it again should show the original expression is a tautology, which I believe is what you want to prove.

Answer 2

Using distributivity,

$\neg A \bigvee \neg((\neg B \bigwedge \neg A) \bigvee (\neg B \bigwedge B))$ $\equiv \neg A \bigvee \neg (\neg B \bigwedge \neg A)$ $\equiv \neg A \bigvee (B \bigvee A)$ $\equiv \neg A \bigvee A$

as required.