There is a problem that asks if an ice cube is dropped into nitrogen (boiling point of nitrogen is 77K), how much nitrogen will evaporate?

Question

heat lost by ice = heat gained by nitrogen

But by is the equation

m(ice)*c(ice)*deltaT = m(nitrogen)*L(vaporation) ?

Why isn't it

m(ice)*c(ice)*deltaT + m(ice)*L(ice) = m(nitrogen)*L(vaporation)

because aren't you melting the ice into water too?

thanks for the help.