Referencing a mathematical result already obtained
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.