Conjunction Elimination and Introduction
Disjunction Elimination and Introduction
Conditional Elimination and Introduction
Negation Elimination and Introduction
Biconditional Elimination and Introduction

Yah, clarification is good. I need to see exactly what I'm allowed to use so I don't slip in details which aren't allowed.

Now, when you say you want to prove it's a theorem, I presume you want to deduce it from the rules of deduction? (As oppose to, say, proving that it evaluates to true for any truth assignment)

Well, by definition in my text, a sentence P of sentential logic is a theorem in sentential derivation if and only if P is derivable in sentential derivation from the empty set.

So by making certain assumptions, can you derive the sentence P using the rules of sentential derivation?

Here is the text that we are using in my philosophy class of symbolic logic just in case you are interested.

I think I follow: It seems like you are doing two negation elimination claims with a negation introduction claim as well.

Is this what you are doing?

Assume ~(P v ~P)
__Assume P
__Then P v ~P (by line 2: disjunction introduction).
__But ~(P v ~P) (by line 1: repetition).
Therefore ~P (lines 2-4: negation introduction).
__Assume ~P
__Then P v ~P (by line 6: disjunction introduction).
__But ~(P v ~P) (by line 1: repetition).
Therefore P (by lines 6-8: negation elimination).
Therefore (P v ~P) (by lines 5,9: negation elimination).

I knew that I had to arrive at a contradiction somewhere to get the conclusion, but I never thought of deriving ~P and P through contradiction. In fact, even after you showed my your proof I had a bit of a time following what you were doing. But I think I got it now.