It really isn't my logic (and unless I am totally wrong/ignorant about this, this is how it works):
You sign on to the internet using your modem which connects to the interet via your ISP (in this case AT&T).
Once on the internet, obviously you can access whatever you want: Yahoo Mail, or any other site.
Using Yahoo Mail from a web browser means you are using Yahoo's mail servers. They have nothing to do with your ISP's mail servers.
Sending Mail from your computer - not a web browser - will connect you to your ISP's (AT&T) mail servers.
And here are copies of:
the headers from an email I sent from my Yahoo email account via my web browser (not using Mail):
Received: from [71.222.39.137] by web120205.mail.ne1.yahoo.com via HTTP; Thu, 25 Oct 2012 08:30:20 PDT
X-Mailer: YahooMailWebService/0.8.123.450
xxxxxxxxxxx@webxxxxx.mail.ne1.yahoo.com>
xxxx@webmail-xxxx.sysops.aol.com>
Message-ID: xxxxx.xxxxxx@webxxxx.mail.ne1.yahoo.com>
(the "xxx" represent long strings of numbers/letters). As you can see the email was sent from Yahoo's mail server (to someone with an AOL email address). There is absolutely no mention of my ISP's mail servers (because they aren't involved at this point at all).
Now here is a header from an email I sent from my Mac Mail app:
X-Smtp-Server: smtp.centurylink.net:xxxxxx@xxxxenturylink.net
Message-Id: <xxxxx@xxxnturylink.net>
References: xxxxxx@xxxenturylink.net>
xxxxxx@xxnturylink.net>
As you see here, the mail server is centurylink, which is my ISP; so emails originating from my computer/Mail app will go through my ISP's mail servers.
If I am wrong, hopefully someone will chime in with the correct information.
FWIW, above "links" really aren't links, they do not actually work (no idea why it would show up as links except that it contains an '@').