If the mechanic shop who sold me and install a used transmission gave me a 30 day warranty . The transmission is still slipping

The short answer is yes. A warrantor is obligated to make adequate repairs under the warranty so that the defect is resolved within a reasonable time frame or number of repair attempts. So long that the defects that you are experiencing now were reported during the time that the warranty was in force you are covered and it is the warrantors obligation to make adequate repairs even if that takes you beyond the actual warranty period. However, if the repair facility does not offer you a refund or to properly repair the transmission you will be reqired to file a case in small claims court. If that's the route you take I would advise retaining an attorney if you can find one that will take the case and wait to receive his or her fees directly from the repair shop after a victory. If you cannot find one who will work under this arrangement you can go it alone but you will probably need to retain a transmission expert to call on your behalf if the matter goes to trial.

Mr. Bolliger is quite correct in his assessment. Keep in mind too that the shop itself may have some type of warranty from the transmission provider, so, the only thing it would be out is labor to replace your used one with another one. You've done the right thing by continuing to give them an opportunity to fix. The rest is up to them. Should they not do the right thing, a good consumer attorney should be able to help. Best of luck with it.