B737-700 pilot reported Captain's air speed and altitude information became erroneous in flight with icing conditions present. The flight crew switched to First Officer's instruments and performed a diversion and a precautionary landing.

Narrative

Aircraft X scheduled to operate ZZZ1 to ZZZ2. Push; engine start; taxi; and takeoff were uneventful. On climb out; aircraft entered IMC and icing conditions; and Captain turned on engine anti-ice. At approximately 16;000 ft. MSL; Crew observed 'IAS DISAGREE' and 'ALT DISAGREE' on the Primary Flight Displays; followed very shortly by Master Caution; ENG caution indicator; and both EEC 'ALTN' lights illuminated. Crew noted the Captain's airspeed to be significantly lower than the First Officer's and standby airspeed; so Captain transferred aircraft control to First Officer.Crew completed the Airspeed Unreliable memory items; and then coordinated with ATC to level at FL190 and maintain current heading to continue troubleshooting. After leveling off; Captain's airspeed showed approximately 100 knots below First Officer's and standby airspeed. Captain's altimeter showed approximately 300 ft. higher than First Officer's and standby altimeter. Crew ran the Airspeed Unreliable Checklist from the Quick Reference Checklist and Quick Reference Handbook; up to deferred items; and requested priority handling with ATC. Crew determined the First Officer's airspeed and altitude were reliable; so in accordance with the Quick Reference Handbook; the First Officer re-engaged Flight Director and autopilot.After determining ZZZ3 as a suitable divert field; Crew informed ATC and Dispatch of intention to divert; and then informed Flight Attendants and Passengers of the situation. Crew then ran the EEC Alternate Mode Checklist from the Quick Reference Handbook; placing both engines in hard alternate mode. Crew also completed the diversion checklist and the deferred items from the Airspeed Unreliable Checklist. On arrival; below approximately 10;000 ft. and in VMC; Crew observed Captain's airspeed and altitude to be indicating correctly. First Officer flew an uneventful visual approach to Runway XXL and overweight landing at ZZZ3 (LGW 131;000 pounds). Taxi in to Gate X and shutdown were uneventful.