Extrapolating an argument from Quine's classic "Two Dogmas of Empiricism":

Observe first that claims by themselves do not have predictive content. Take the claim 'The chair in my office is black'. By itself, this claim does not tell us anything about future experience. You need a huge background body of claims to make predictions: e.g. once and only once you also believe 'Objects tend to stay where they are unless moved', 'No one is likely to move my chair', 'Black objects react to light in such and such ways', 'Light waves propagate in such and such ways', 'The propagation is described by such and such mathematics', 'Such and such mathematics is justified by such and such axioms', 'Those axioms are true', etc, can you predict things like 'When I step into my office I will have so and so experiences (seeing black-chair-like things)'. Presumably though, the way that we come a posteriori to know a claim is that we verify its predictions. By the above sketched account (the Quine-Duhem thesis) we verify not claims by themselves but entire bodies of claims.

Note further that mathematics, logic, etc. (the traditional candidates for a priori knowledge) are an essential part of our body of claims. Hence they too get verified by future predictions. This means that we can come to know them a posteriori (by verifying the predictions of the body of claims they belong to). To write the argument up it goes like:

1. Claims are verified as a group not singly.
2. A sufficiently verified and true group of claims has all of its members knowable a posteriori.
3. Putative examples of a priori knowledge (mathematics, logic, etc.) are part of such a verified group.
4. Therefore, Putative examples of a priori knowledge are knowable a posteriori.

Once we have this conclusion we can reject the claim that those claims are or can be known a priori along the following lines:

1. The mechanisms that philosophers have proposed for coming to know claims a priori (such as rational intuition or ratiocination) are obscure or question begging.
2. If there are two competing accounts about the methods we come to know a class of claims, one of which is obscure and the other clear, we should accept the latter and reject the former.
3. The Quinean answer of how we come to know, e.g., logic and mathematics is clear.
4. Therefore, we should accept the Quinean answer and reject that there are mechanisms for knowing claims a priori (i.e. we should reject that there is a priori knowledge).

Hope this helps.