AN ACR B737-500 FLC HAS A POTENTIAL CONFLICT WITH AN ACR B727 WHILE MAINTAINING VISUAL SEPARATION FROM YET ANOTHER ACR B727.

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AFTER TKOF FROM RWY 16L SLC; TURNING TO 280 DEG HDG; CLRED TO CLB 11000 FT. ON HDG 280 DEGS AT ABOUT 6500 FT; ASKED BY DEP CTL IF WE HAD ACR Y; A B727; IN SIGHT AT 7000 FT. WE RPTED THAT ACFT OFF OUR NOSE ABOUT 3-5 MI IN SIGHT; AND CONFIRMED THE ALT ON TCASII. WE WERE THEN CLRED TO CLB TO FL230 IN A R TURN DIRECT TO SLC VOR; AND TOLD TO KEEP ACR Y IN SIGHT 'MAINTAIN VISUAL SEPARATION.' WE STARTED THE TURN AND CLB; I WAS HEADS DOWN ON THE FMC WHEN I NOTICED TFC ON TCASII L SIDE; LESS THAN 1 NM; DSNDING FROM 11200 FT. I LOOKED OUT THE CAPT'S L WINDOW AND SAW AN ACR Z B727 CLOSE ABOARD (3000 FT) ON A HDG OF ABOUT 330 DEGS. WE WERE AT CLOSEST POINT OF APCH ABOUT THE TIME I SAW HIM VISUALLY AND OUR NORMAL TURN WAS INCREASING SEPARATION. I CALLED DEP AND QUESTIONED THEM ABOUT THE TFC. HE SAID THAT THEY HAD NO ACR Z ACFT IN THE AREA. I DESCRIBED THE TFC AND HE RESPONDED THAT THEY ASSUMED WE HAD THAT TFC IN SIGHT. FACTORS: ATC DID DIRECT US TO MAINTAIN SEPARATION FROM ACR Y. THIS WAS NOT THE ACFT IN CONFLICT. ATC ASSUMED WE WERE LOOKING AT ACR Z. OVERTASKED? LOSS OF SITUATIONAL AWARENESS? EXPECTED RESPONSE FROM CREW? THIS WAS A TCASII SAVE. I WOULD NOT HAVE THOUGHT OF LOOKING FOR A BELLY CHK IF I HAD NOT NOTICED THE CONFLICT ON TCASII (WE NEVER HAD AN RA; ONLY TFC FOR 5-10 SECONDS). WE WRE CLOSE ENOUGH THAT LEVELING OUR WINGS WHEN WE SAW THE CONFLICT WOULD HAVE CAUSED A COLLISION IN SHORT ORDER. I ASSUME ACR Z NEVER SAW US.