Extension of propositional logic:

Away from proving  theorems directly, and the other main use for rewrite rules is to prepare a statement just for use before we search for the proof when described it in the next lecture. Because there some automated deduction techniques require a statement to be in a particular format so there in these cases we can use a set of rewrite rules to convert the sentence that we really want to prove in a logically equivalent one that is in the correct format. 

Lower than some common equivalences that automated theorem proves can be need as rewrite rules. Memorize there that the rules can be read both ways, except that in practice either i) a loop-check is employed or ii) only one direction is used. Remember there also that these are true of sentences in propositional logic, than they can also be used for rewriting sentences in first-order logic, that is just an extension of propositional logic.