Boolean Simplification: Identifying a rule

Question

I'm in the process of minimizing a boolean equation, and I've gotten it into the following form:

$$\lnot B \lor (B \land \lnot C) \lor C$$

Just by looking at it, I can tell that this is always TRUE. The complements of the center term are covered by the other two. However, I can't quite prove it formally.

I can split the middle term using indempotency:

$$\lnot B \lor (B \land \lnot C) \lor (B \land \lnot C) \lor C$$

Then (again, intuitively) I can tell that the two left terms simplify to: $\lnot B \land \lnot C$

What is the rule (or rules) that can get the left two terms into this form? Once I know that rule, I can formally solve the equation like so:

$$ \lnot B \lor \lnot C \lor C\lor B$$ $$(\lnot B \lor B) \lor (\lnot C \lor C)$$ $$ 1$$

Any help would be greatly appreciated!

Answer 1

$$\lnot B \lor (B \land \lnot C) \lor C$$ rearrange $$(\lnot B \lor C) \lor (B \land \lnot C)$$

Let's apply the De Morgan rule here quick:

$$(\lnot B \lor C) \lor \lnot(\lnot B \lor C)$$

I think it is called the "law of excluded middle" that $ A\lor \lnot A$ is always true, and that's what you have here for $ A = (\lnot B \lor C) $.

Answer 2

$$\lnot B \lor (B \land \lnot C) \lor C$$ $$(\lnot B \lor B) \land (\lnot B \lor\lnot C) \lor C$$ $$T \land (\lnot B \lor\lnot C) \lor C$$ $$(\lnot B \lor\lnot C) \lor C$$ $$\lnot B \lor (\lnot C \lor C)$$ $$\lnot B \lor T$$ $$T$$