Hebrews 7:18 states the annulling of the former commandment as the consequence of the priesthood change, following logically from here.

Hebrews 8:13 declares the old covenant obsolete, the direct result of the law change necessitated by the priesthood change here.

Jeremiah 31:31-34 prophesies a new covenant replacing the old, providing the prophetic basis for the law change required by the priesthood change.

Acts 6:14 quotes Stephen being accused of saying Jesus will change Mosaic customs, directly paralleling the necessary law change here.

Isaiah 66:21 prophesies God taking foreigners as priests, implying a change in priesthood order that supports the law change here.