Two-player first-price auction with resale
Let's look at this with backward induction. Let $v_{i}$ be my valuation and $v_{-i}$ be your valuation. Suppose I've won the item. Then $v_{i} \geq v_{-i}$. If I sell you the item at price $v_{-i}$, then my utility is $v_{-i} - b_{i} \leq v_{i} - b_{i}$, where $b_{i}$ is my bid according to the symmetric equilibrium bidding strategy (which may not necessarily be $\beta$). Notice that I do not gain $v_{i}$ by selling you the item, as I do not possess it.

We see that I will never have incentive to sell you the item. So I should bid as if I'm not going to sell you the item. We have an equilibrium bidding strategy for this anyways, given by the first price auction.