AN ACR MLG RESPONDING TO A TCASII ALERT ON VFR TFC; CLBED TO WITHIN 1 MI AND 500 FT OF AN IFR CORP ACFT.

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I WAS WORKING CORP JET X; A DEP OFF CGX WBOUND. I CLBED THE CORP JET X TO 12000 FT. HE WAS LEVEL AT 12000 FT ABOUT ORD 180/20 ON A 280 DEG HDG. ACR Y WAS AN ORD ARR AT 11000 FT COMING FROM THE SW. A KNOWN VFR ACFT Z (ANOTHER ACFT THAT APCH WAS TALKING TO) WAS AT 10500 FT IN THE SAME VICINITY. ORD ARR CTLR POINTED OUT VFR TFC TO ACR Y. I SHIPPED CORP JET X TO ZAU FREQ 127.07. ACR Y STARTED TO CLB BECAUSE OF TCASII ALERT OFF VFR ACFT. ACR Y WAS CLBING UP INTO CORP JET X. I TRIED TO COORDINATE WITH ARR CTLR AS HE WAS ALSO TRYING TO RESOLVE CONFLICTION. IT APPEARED THAT VFR ACFT WAS W OF ACR Y AND NO FACTOR BEFORE ACR Y RESPONDED TO HIS TCASII ALERT. SUPPLEMENTAL INFO FROM ACN 265103: WE WERE LEVEL AT 11000 FT. ORD APCH CTL HAD US ON A HDG; VECTORING FOR RWY 27L AT ORD. APCH CALLED VFR TFC AT 10500 FT AT OUR 11 O'CLOCK POS. I TOLD ATC NO CONTACT. TCASII SHOWED TFC 500 FT BELOW US. THE TFC WAS WITHIN 1 MI WHEN TCASII TOLD US TO MONITOR VERT SPD. TCASII THEN SHOWED TFC 400 FT BELOW US AND CLBING. TCASII TOLD US TO CLB. WE STARTED CLBING AND CALLED APCH AND TOLD THEM WE WERE CLBING OUT OF 11000 FT BECAUSE OF A TCASII ALERT. TCASII THEN TOLD US TO MONITOR VERT SPD. TCASII THEN TOLD US TO REDUCE CLB. WE REDUCED. AT 11500 FT WE HAD VISUAL CONTACT WITH A CORPORATE JET. HE WAS ABOUT 1000 FT ABOVE US. APCH THEN TOLD US OF THE CORPORATE JET CONFLICT. TCASII CALLED CLR OF CONFLICT. WE STARTED DSNDING BACK TO 11000 FT AND LEVELED. IF THE VFR TFC AT 10500 FT DIDN'T CLB THE SIT WOULD NEVER HAVE HAPPENED.