Exactly, so if inflation increases, RPI will go down all other things held constant.

Personal income also increases with inflation, unless of course you are in a recession, where "all other things held constant".

This is circular reasoning. It's also wrong as a general claim. Personal income, at least as traditionally measured, fell for a large number of US workers over a 35+ year period during which inflation was non-zero and yet there was not a recession covering the entire period.