I understand that a decrease in prevailing market interest rate causes a downward pressure on the bond yield and, equivalently, an upward pressure on the bond price. But what if a decrease in interest rate leads the investors to anticipate inflation? Wouldn't the anticipation of inflation cause the bond to be worth less and subsequently exert a downward pressure on the bond price?
The price of a bond is tied to the interest rate (yield) of that bond by a simple formula, so the effect of one on the other is fully predictable. The relation between prices and/or yields of different bonds -- e.g., with different maturities or credit risks -- is much more complex and can involve secondary effects of the kind you mention. These are measured by yield curves and yield spreads.
If the "prevailing market interest rate" you refer to is a short-term rate like the Federal Funds Rate, then indeed it can happen that a rate cut leads to an increase in long-term bond yields (steepening the yield curve) due to inflation expectations. Or vice versa, short-term rate hikes may not increase long-term rates (the yield curve may flatten) -- this was Alan Greenspan's famous conundrum in the mid-2000s.
If rates are increasing and bond prices are declining, that makes older bonds less attractive because newer bonds offer a higher rate. In order to be competitive, the price of the older bonds must decline otherwise no one would buy them. Conversely, if rates are declining, investors/traders are wiling to pay more for the older bonds because they offer a higher yield than current bonds.
This process happens as rates change as well as when investors/traders anticipate that rates are going to change.
Bonds are sold ('offered') with a defined interest rate, which doesn't change afterwards.
Imagine you bought a 1000$ - 3% bond, and the market interest goes down below that - your bond is suddenly better than other investments on the market, so demand goes up, and people pay you more than the 1000$ to have it. On the other hand, if market interest goes up, over the 3%, your bond is now a poor investment, as you could get more than 3% on the market - so people won't pay you the full 1000$ for the bond.
As the current market interest rate and your bond's info are known, offer and demand adjust your bond's price perfectly so that a buyer of your bond makes exactly the market interest rate - not more and not less.