Streaming Replication - changing IP addresses

Greetings.
We have streaming replication set up on two servers that are on our local
network using their external/public IP addresses. We are switching internet
providers and need to change the external/public IP addresses of both
servers. I'm not sure how to go about this correctly.

1. Our local network IP addresses will not be changing, is there any
reason we could/should not use our local network IP addresses in the
streaming replication process?
2. What is the best route to make these changes? Is it as simple as
turning postgres off on both servers, changing the IP settings (master -
postgres.conf and hba.conf and slave - recovery.conf files to the local
network IPs) and starting the service up again? Can anyone point me to an
example?

Any tips, suggestions, help you could provide would be greatly appreciated.

Thanks in advance for your time!

Dara