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If the ice is not already at 0°C, then you must heat the ice in addition to melting it (using ΔH of fusion). Therefore, we use the specific heat of ice to heat it to 0°C, then we melt it, then we heat the now liquid water using the heat capacity of water. Likewise, with the water, we use its specific heat to find how much temperature drops.
The energy gained by the ice is equal to the energy lost by the water. However, if the water has enough energy to heat the ice to 0 degrees C and melt it, then to get the melted ice to the final temperature you will need to use the specific heat of liquid water even though it would be on the same side of the equation as total energy change for the ice.
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