Not Gay-Lussac's Law

Crushing a can by heating it, sealing it and cooling it rapidly is an excellent demonstration of the existance of atmospheric pressure. Many youtube videos describe the crushed can as an illustration of either Charles' Law or Gay-Lussac's Law. I don't agree.

There are many variations on the theme, but the basic experiment involves heating a small amount of water inside of a metal container (usualy a soda can). The can is then sealed and cooled rapidly. This results in a drop in pressure inside of the can. The drop in pressure inside of the can allows the higher external pressure from the atmosphere to crush the can.

Why does this setup not demonstrate Charles' or Gay Lussac's Laws?