That would create a problem in the U.S. Every RNAV IAP states "DME/DME RNP 0.30 not authorized."
Further, every RNAV SID and STAR must be evaluated for critical DME updating for the old, pre-GPS birds. If there is critical DME then the SID or STAR is not authorized for non-GPS platforms when that critical DME is inoperative.
The pecking order of all modern air carrier and high-end biz jet FMS navigators is:
GPS
Blended IRU position
DME/DME
VOR/DME
VOR/VOR
Further, to fly most RNAV procedures, only GPS is required; IRUs, DME/DME, VOR/DME, and VOR/VOR are not required, in fact, except for IRU(s) not authorized, to use the procedures.
In the case of RNP AR at least one IRU is required in addition to two GPSes and two FMSes. The exception is RNP AR of not less than 0.30 in any segment to be flown and RNP of less than 1.0 is not required for the missed approach. In that case, no IRU is required.
Not exatly true in all cases.
For example the Primus Epic system goes:
Blended GPS
single GPS
then whoever has the lowest EPU from:
DME/DME
VOR/DME
IRS.
PBN specifications don't match the descriptions you give. For example RNAV 1 can be GPS or DME/DME or DME/DME/IRS. Each state can declare which is acceptable. If DME/DME or DME/DME/IRS only then the critical DME assesment is required. If GPS then the older equipment won't be RNAV 1 certified.
In China for RNAV 1 arrivals and departures it's a requirement to inhibit VOR/DME fixing as it doesn't meet RNAV 1 specifications.