No individual shall receive any exemption to which he is entitled by any one of subdivisions (19), (20), (22), (23), (25), (26) and (28) of § 12-81 or § 12-82 until he has proved his right to such exemption in accordance with the provisions of sections 12-93 and 12-94, together with such further proof as is necessary under the provisions of any of said sections. Exemptions so proved by residents shall take effect on the next succeeding assessment day, provided individuals entitled to an exemption under the provisions of subdivision (20) of § 12-81 may prove such right at any time before the expiration of the time limited by law for the board of assessment appeals of the town wherein the exemption is claimed to complete its duties and such exemption shall take effect on the assessment day next preceding the date of the proof thereof. For purposes of any tax payable in accordance with the provisions of § 12-71b, any such exemption referred to in this section shall take effect on the first day of January next following the date on which the right to such exemption has been proved.