Referencing a mathematical result already obtained I was working(in maths) for some months on a paper. After I got the results, I found out that some of those were already obtained in another paper that was not very related to my topic. In fact, in that paper, the result is stated as of 'minor' importance because the main purpose of the paper was quite different. It is the technique that he used to prove this, the one that I rediscovered and used in my paper to prove this result and others in the same domain. Due to this, I didn't have a reason to search there in the first place. Although I have other results in my paper, this one was one of the main ones. Of course, I want to cite the old paper.  Hence my questions: 


Is it worth publishing this with a proof?
Assume now that I do publish it with proof. Is a phrase like: "Although we arrive at this result independently, it turns out that this was already obtained in [Ref]" acceptable in the paper? 
Is there a more correct way than the phrase above to cite the old paper?
Since your proof is essentially the same as the proof in the literature, you have to state a good reason for including it anyway. Something like "although this argument has already appeared in [citation], we include a proof to demonstrate its connection with our other results, and also for completeness." But the longer the proof is compared to the length of your paper, the less valid a justification like this will be. If this proof takes up more than say 25% of your paper*, I recommend removing it, and simply mentioning the relationship between that theorem and your other results. There's nothing wrong with short papers, if the results are good.

Either way, understand that your paper will be judged on what's new in it, and write your introduction accordingly.

*People may disagree with this number in either direction, but it gives you a rough idea. Certainly 50% would be too much, and 10% is probably fine.