Praise be to Allah; and may His blessings and peace be upon our Prophet Mohammad and upon his Family and Companions.
It isn`t a condition that a mortgaged item should be owned by a debtor because a mortgage is a security for the debt due on him, and since it is permissible for the owner of the item to mortgage it in return for the debt due on him, then it is also permissible for him to mortgage it as a security for the debt due on someone else, provided that he does so voluntarily and without charging any profit. Al-Imam Ashirbini said: “ It is permissible for a debtor to borrow an item as a mortgage for his debt because this method secures the money due on him to the creditor. The most preponderant opinion of the scholars is that a mortgage is a security provided by the owner of the item in return for a debt because as long as he enjoys the authority to guarantee the debt of someone else, then he enjoys the right to pay it off in case the debtor failed to do so. [Moghni Al-Mohtajj(3/52)].
Therefore, it is conditioned that a debtor tells the guarantor as to the exact amount of the debt, its kind, its duration and the name of the creditor so as for the contract to become binding to him (guarantor).
If the creditor failed to collect his money from the debtor, then he should claim it from the guarantor because the latter may pay him, if not, the debtor, or the guarantor sells the mortgaged item and pays the debtor an amount equal to its value. Al-Imam An-Nawawi said: “ Once the debt`s repayment has become due, the owner of the mortgaged item is consulted as to selling it in case the debtor failed to pay the creditor. “ {Minhajj At-Talibeen(115)}.
In conclusion, it is permissible for the first seller, in the above question, to collect the installments of the debt owed to him after selling the mortgaged car in case the debtor stopped paying him. And Allah knows best.