Can I fund existing IRA with brokerage cash via bank

Question on funding an existing IRA (rolled-over 401K from a previous employer).

I understand a brokerage account isn't a qualified retirement plan so I cannot do a direct transfer, but I would still like to use its cash to fund the IRA with a tax deductible 2020 contribution.

Is it allowed to: 

Step 1 Move cash from the Brokerage to my bank account;

Step 2 Move that cash from the bank account to the IRA?

I think "yes" but I can't find a definitive answer and the last thing I want is get into trouble with a three letter acronym government agency...  Thanks much in advance!