No, you're just changing the account custodian. The 5 year rule would apply if you're still working and therefore had earned income AND deposited it in an entirely new Roth IRA account. And why anybody would do that I cannot say, because it wouldn't ...

I’m pretty computer literate but unlike my economist friend, I continue to keep paper & pencil ledgers of my checking and savings transactions.  Like him, I do trust the bank and other institutions to do this for me—even better than me actually.  BUT...