This is ultimately a judgement call for you to make. However, you having been together for three years is not a good reason to stay with someone in and of itself. Three years is a long relationship in college terms, but it is a very short relationship in the context of the rest of your lives. You cannot maintain a long term relationship without trust. The reality is that if you cannot trust him to remain faithful to you, the relationship cannot and will not work, and it is far better to break it off now than later down the line after you've lived together significantly longer, are married and/or have had children, when the breakup will be immeasurably more painful and difficult.
In a minority of cases, relationships can recover from someone cheating, but that very much requires the person who has cheated to be honest and up front about their behaviour, recognise the significance of it, and actively work on the factors that led to it so that they can rebuild trust and their partner can be satisfied that it will not happen again. That is usually just not possible, and here it seems as though your boyfriend initially lied about cheating on you, and then played down its significance when shown proof of it. The dishonesty and minimising his behaviour are both massive red flags in my view. Unless there is more detail here that we're missing which makes the situation retrievable, I think it's a very easy decision to break up with him. You deserve better.