Let's take a look at I.D. from the homework.  That argument looks
like this:

           P & Q
           -----------
           P & (Q v R)

        None of our rules of inference will give us our conclusion
directly.  It looks like addition might, but it won't.  That is because
each rule of inference applies only to whole lines, not to parts of lines.
If we applied addition to our premiss, we would get ([P & Q] v R), which is
not the conclusion we want.  Nevertheless, we can derive (P & [Q v R]) from
(P & Q).  By simplification, we can get P from (P & Q), and by
simplification again, we can get Q from (P & Q).  What simplification
allows us to do is derive from any conjunction either of its conjuncts.  P
and Q are the conjuncts of (P & Q), so we can derive each from it.  Now
that we have derived Q from (P & Q), we can use it as another premiss.  By
applying the rule of addition to Q, we can get (Q v R).  That is now
another premiss we can work with.  With the rule of conjunction, we can
derive (P & [Q v R]) from our premises, P and (Q v R).  This rule lets us
derive from any two premises the conjunction of those two premisses. Thus,
we have derived (P & [Q v R]) from (P & Q).

        In the more formal way that you will soon learn, our derivation
looks like this:

               1.   P & Q                   Premiss
               2.   P                       1, Simp.
               3.   Q                       1, Simp.
               4.   Q v R                   3, Add.
               5.   P & (Q v R)             2, 4, Conj.

        Each line contains a symbolic expression.  The text at the right
states our justification for including each line in the derivation.  The
first line contains our premiss, so it says "Premiss."  The subsequent
lines tell which lines their expressions were derived from, and according
to which rules.  For instance, line 5 was derived from lines 2 and 4 by the
rule of conjunction.