B737 FLT CREW RECEIVES A TCASII RA DURING CLB TO ALT AFTER DEPARTING SLC.

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DURING A FLT FROM SLC TO GEG; A TCASII RA WAS ENCOUNTERED. THE AUTOPLT WITH VNAV AND LNAV WAS ENGAGED. ACFT WAS AT 310 KIAS; VMC; LEVEL AT 17000 FT WITH ZLC. AN ACR Y WAS AHEAD AND ABOVE AT FL220. THE CAPT AND I HEARD ACR Y REQUEST AND RECEIVE A CLB TO FL240 FOR TURB. WE WERE THEN CLRED TO CLB AND MAINTAIN FL230. DURING THE CLB; APPROX 1/2 NM BEHIND ACR Y; WE RECEIVED A TCASII RA TO LEVEL OFF/DSND. I IMMEDIATELY DISCONNECTED THE AUTOPLT AND INITIATED A DSCNT BACK TO FL210 IN ACCORDANCE WITH TCASII COMMANDS. WE PASSED BELOW AND TO THE L OF ACR Y. UPON HEARING TCASII ADVISE 'CLR OF CONFLICT;' I INITIATED A SLOW CLB. APCHING FL220; ACR Y QUERIED SLC CTR ABOUT THE RA; SO WE DSNDED BACK TO FL210. DURING THE RA; WE ADVISED SLC CTR OF THE ALERT; OUR RETURN TO FL210; AND THAT WE HAD ACR Y IN VISUAL CONTACT. (THE CTLR SOUNDED CONFUSED AS HE THOUGHT HE HAD CLRED US ONLY TO FL210.) UPON REACHING SPOKANE (GEG); THE CAPT CALLED THE SLC CTR SUPVR AND WAS TOLD THAT HIS CTLR HAD CLRED US TO FL230 (AFTER REVIEWING THE TAPES). HE SAID IT WAS CTLR ERROR AND THAT HE WAS GLAD WE HAD MAINTAINED VISUAL CONTACT WITH ACR Y AND FOLLOWED TCASII COMMANDS. HE FURTHER SAID NO ACTION WAS GOING TO BE INITIATED AGAINST US. THIS WAS LEG 3 ON DAY 2 OF THE ROTATION. FATIGUE WAS A CONTRIBUTING FACTOR. ALSO; ALTHOUGH ALL TCASII COMMANDS WERE FOLLOWED AND VISUAL CONTACT MAINTAINED; A MORE AGGRESSIVE AND EARLIER DEV MIGHT HAVE NEGATED THE TCASII RA.