c100driver and 172_driver, my question was specific to flying P-RNAV STARs and SIDs with an aircraft that is equipped to fly RNAV/GNSS but is temporarily prohibited from flying DME/DME P-RNAV due to the MEL. In those circumstances, I'm not sure the ANP would reveal anything.
I do agree with Skyjob's idea of ascertaining a RAIM check, or you might be relying on a GPS position that is possibly corrupt. Apart from that, if the performance of the navigation system as a whole is sufficient for RNAV1 I cannot see any safety implications? We rely on GPS for RNAV approaches (RNP0.3), post RAIM check, so why not PRNAV.
The MEL is in my opinion not a straight forward document and various interpretations are often needed to comply. The note of MEL 34-13 (.."Both required for RNAV1 (PRNAV) operations when the procedures are based on DME/DME"..) is impossible to answer for the line pilot, sitting on the flight deck facing the MEL item.
That's my reasoning to come to my conclusion that if ANP is less than RNP1.0, I would not object to using the procedure.