B737 Captain reported First Officer's Airspeed failed with IAS Disagree was displayed on both PFDs in cruise. The flight crew ran the QRH and check lists; then both EECs went into Soft Alternate Mode and Speed Trim Failed. The flight crew requested priority handling and landed at destination airport.

Narrative

At 360 the FO's (First Officer) airspeed indication failed. IAS Disagree was displayed on both PFDs. A/T Limit was displayed on UD. I was PF (Pilot Flying) at the time worked together and agreed it was the FO's A/S that had failed. We transferred controls and accomplished A/S Unreliable QRH. After completing the QRH; both EECs went into soft Alternate Mode and Speed Trim failed. Shortly after this the Standby Attitude Indicator rolled right 30 degrees and went nose low 10 degrees. The FO's Altitude Indicator indicated 500 ft. off from CA (Captain) and Standby. we fired off a quick ACARS to Dispatch notifying of Systems Failures and situation. Flight Attendants were briefed.After we transferred controls; I became PF and took over ATC communications. Our first communication to ATC was we were experiencing instrument problems our Airspeed Indicators were failing as well as standby indications. ATC did not understand this and instructed us to turn left and speed up for spacing into ZZZ. I decided to request priority handling as the loss of A/S indications made it necessary. I requested direct ZZZ and lower altitude to get out of the clouds. We were cleared direct ZZZ and 10;000 ft. Once the workload slowed down and QRH accomplished; we both agreed to have ARFF standing by out of caution. We advised ATC we wanted ILS XX and the approach to landing was without event. Once clear of runway I told ATC no assistance would be needed from fire rescue. We taxied to gate and deplaned normal via jet way. I then contacted Dispatch; Maintenance Control and Operations as required for debrief. Appropriate ELB entries were entered and a more thorough brief via phone with Maintenance Control was accomplished.I did not alert the passengers to the situation. Both CA and FO agreed after accomplishing all QRH items we had working instruments on CA side and if nothing changes the landing would be normal and there was no need to stress the passengers.