Originally Posted by Kuan

When I went to school we had to first prove DeMorgan's law in order to use it. In any case, first step remains the same. Prove it going one way, then prove it going the other. You just have to figure out the equivalents just like in algebra.

Here's a tip. Anything inside parantheses, for example (p -> r) v (q -> r) you can try to look at it as if everything inside the parantheses is just one. So think:

(p -> r) becomes the P in DeMorgan's law and (q -> r) becomes the Q in Demorgan's law.

Demorgan's law is -P or -Q is equivalent to -(P and Q)

Since P is equivalent to --P (double negation) or if you want to look at it like -notP

-notP or -notQ is equivalent to -(notP and notQ)

Hence

(p -> r) v (q -> r) by DeMorgan's law becomes -(-(p -> r) AND -(q -> r))

You really only need three rules plus negation for any proof. Material implication which changes a conditional to a disjunction and both DeMorgan's laws. That is the sum total of Boolean Logic.