Those symptoms do sound like hypothyroidism and symptoms are the most important indicator; however, additional tests are required. You can get some good info from this link. I suggest reading at least the first two pages and also note the tests suggested in no. 6 on page 2. You can also share the paper with your doctor to try and get treated clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
There is no way of being sure from just those test results. What, if any, symptoms do you have? Also, if the doctor wants to be sure why doesn't he just test for the antibodies of Hashi's? Two tests may be required: Thyroid Peroxidase antibodies (TPO ab) for sure, and Thyroglobulin antibodies (TG ab) if TPO ab test is negative.