There’s the law of one price.

When interest rates go up, older bonds with lower yields go down in price and trade at discounted prices so that they have a higher effective yield to maturity that matches the higher yield of newer bonds.

When interest rates go down, bond prices go up, for the opposite reason. New bonds aren’t going down in price when interest rates go down; older bonds are constantly matching the yields/prices of newer bonds, and all bonds (old and relatively new) are constantly matching whatever the most recent bond trade was, since the recent auctions and the recent secondary market bond transactions are what set the price.