When can we eliminate disjunction within equivalence?

I'm a first year philosophy student and I've got a question about elimination of disjunction. Actually, I'm not sure if this can be called elimination of disjunction, we hadn't covered this yet, but it's been bothering me so I had to ask! It seems that within equivalence, for example, p ^ q <=> -(-p v -q), we can take both sides and add v p to them, and that truth of equivalence itself wouldn't change. So, (p ^ q) v p <=> (-(-p v -q)) v p is tautology as well. Naturally, we can remove what we added and tautology still remains.

This doesn't seem to always be the case. While it seems that we can always add the disjuntion on both sides, we can't always remove it. If we had for example: -p v p <=> (p => q) v p And we removed the disjunction with p, -p <=> p=>q would remain, which is false. Have I made a fundamental mistake when I said we can add a disjunction on both sides? When is it alright to remove it?

Thanks!

Something seems off with the system of inference that you are using, specifically in the idea of adding to both sides. The problem is that claims in logic are not generally taken to be two sided like an equation.

p ^ q <=> -(-p v -q) = DeMorgan's Rule

That's an available inference rule in many systems.

-p v p is also special because it is axiomatic for systems that occur under the normal rules of logic.

Normally, you are always allowed to add a disjunction to a claim (not to both sides of a claim -- but with biconditionals, there's a way to do so using conditional proofs).

1. P & Q A
2. (P & Q ) v C vI1

But to take it away, you need to prove the negation of one side.

1. P & Q A
2. ~C A
3. (P& Q) v C vI1
4. P & Q vE3 ,2

What you are saying about taking it away after you add it only follows from the fact that you can just repeat your assumption as a claim later in your argument. (You cannot normally just eliminate a disjunction -- think about it. Either you are tall or you are short. Therefore you are tall. ???)