Is it permissible to purchase a car mortgaged to the bank by paying a portion of its price, with the seller committing to pay the remaining installments, and then finalizing the sale contract on the same day? And what is the appropriate legitimate formula for such a transaction?

If the car is mortgaged to the bank, its owner must inform the bank of his desire to sell it and arrange with the bank the procedures for releasing it from the mortgage, because it is not permissible to sell a mortgaged item without the permission of the mortgagor. If it is released from the mortgage or the bank permits its sale, then it is permissible to purchase it by agreement with its owner.