You will have to Choose, Export as Movie, and export it as an MOV file. This will create a very large Raw Format of an MOV, as there are no compression settings, except the Resolution size. So I would advise you insure you have enough space on your hard drive, for Gigabits of a file. Average can be 16 gigs for just a few minutes.
After which if it does not already pop open, the media encoder will open from there I would create a good format MP4 and at the same time your AVI version.
Once you have a nice clean MP4, I would delete the .mov file, And use the MP4 for any further conversions. Unless you have bags of room for files of great magnitude.
The Export as Movie you describe is how it would be in old versions of Flash Pro. More recent versions have an Export Video option.
But, that's for exporting an FLA. If you have Premiere or After Effects you should be able to import the SWF there, and export video.
Colin is right, my bad, I was writing without looking the option to export to MOV is :
File ---> Export ---> Export Video
And yes, if you have Premier or After Effects you could import the published swf, and even edit the animation again, add effects and music before exporting it to a file type of your needs.