Meteorology Bleg

[kent_allard_jr]

New York is going through another warm spell, and people often respond to these by saying, "man, I can't wait for the rain to come and cool things off!"

I always thought this was a bit of spurious correlation: While the water has a short term cooling effect, rain itself doesn't lead to milder weather; instead the rain is caused by the drop in temperature. Here's what I thought happened:

1. A cold front moves into an area of warm, moist air;
   
2. The warm air is pushed up above the cold air;
   
3. It cools as it rises;
   
4. As it cools, small droplets of water condense;
   
5. which fall as rain.

Is this more or less correct, or am I missing something?