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Let's look at this with backward induction. 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. 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.

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.

Let's look at this with backward induction. 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 $v_{-i} - \beta(v_{i}) \leq v_{i} - \beta(v_{i})$. 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.

Let's look at this with backward induction. 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. 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.

Let's look at this with backward induction. 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 $v_{-i} - \beta(v_{i}) \leq v_{i} - \beta(v_{i})$. Notice that I do not gain $v_{i}$ by selling you the item, as I do not possess it.

Let's look at this with backward induction. 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 $v_{-i} - \beta(v_{i}) \leq v_{i} - \beta(v_{i})$. 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.