Conditional probability in Kaplan, Menzio (2014)
The first / second seller terminology can be confusing here (does it relate to time or to conditionality?). It's safer to focus on a particular seller.

Let the probability that a buyer finds a particular seller, $s$, via single search be $\nu_s$. The probability that a buyer is matched with $s$ and (either before or after) another seller is then (assuming that $\nu_s$ is small so that the probability of finding $s$ twice in at most 2 searches can be ignored):

$$\psi_u(\nu_s\nu+\nu\nu_s)= 2\psi_u\nu_s\nu$$

The probability that a buyer is matched with $s$ via either a single or a double search is (again ignoring the probability, given a double search, of finding $s$ twice):

$$(1-\psi_u)\nu_s + \psi_u(\nu_s + \nu_s)=\nu_s(1+\psi_u)$$

The conditional probability of a buyer being matched with $s$ and another seller, conditional on $s$ being matched with that buyer, is therefore:

$$\frac{2\psi_u\nu_s\nu}{\nu_s(1+\psi_u)}=\frac{2\psi_u\nu}{1+\psi_u}$$

But the latter formula does not depend on $\nu_s$. Hence the same formula applies to any seller.