The melting point of a substance is the temperature at which the solid form's vapor pressure equals the liquid form's vapor pressure. I read that the vapor pressure of something depends ONLY on the temperature. So as the external pressure changes, the substance's vapor pressure should stay the same if temperature is constant. In that case, the melting point should not be affected by external pressure.
I understand that intuitively, melting point increasing with pressure makes sense, as more energy is needed to overcome larger external pressure in order to liquefy. But how can this be explained with the vapor pressure definition?