If you insist on doing all this at the same time (because it is the day the bill is due, of course), you may end up inadvertently transferring money FROM your credit card into your regular checking account, instead of the other way around. And then, not only would the credit card not be paid, you might also get a late fee. And that's in addition to the big fee for the "cash advance" you just gave yourself from the credit card.
This might look like an exciting circus act, like spinning a dozen plates while balancing upside down underwater with hungry sharks, but do not be fooled. It is not fun. Do not try this at home.
Don't ask me how I know.