Soldato
Sounds like you're mistaken (worrying as I expect lots of people assume they have DOC when they don't and are driving other people's cars totally uninsured) - it will specifically state on your insurance certificate that you are also allowed to drive other cars.
Also I have no idea where you got the idea that the other car must be insured - have you really read your documents properly?
1b Driving other cars
If your certificate of motor insurance says so, this policy provides the same cover as above in 1a when you are driving any other motor car as long as you do not own it and it is not hired to you under a hire-purchase or leasing agreement. This cover only applies if:
• there is no other insurance in force which covers the same liability;
• you have the owner’s permission to drive the car;
• the car is registered in and being driven in Great Britain, Northern Ireland, the Republic of Ireland, the Isle of Man or the Channel Islands; and
• you still have your car and it has not been damaged beyond cost-effective repair.
Also I have no idea where you got the idea that the other car must be insured - have you really read your documents properly?