ACR X TCASII TA/RA EVASIVE ACTION TAKEN CLB HAD LTSS SEPARATION FROM CPR Y. SYS ERROR.

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ACR X BEING VECTORED BY CHICAGO APCH CTL (FREQ 119.0). I WAS LEVELING ACFT AT 2700 FT MSL AND WAS GIVEN A TURN FROM AN ASSIGNED HDG OF 360 DEGS TO 310 DEGS TO INTERCEPT RWY 22R LOC. WHILE TURNING; WE RECEIVED A TCASII TA. IT WAS INDICATING AN APCHING ACFT FROM MY SIDE (L SIDE). I WAS LOOKING OUT THE WINDOW TO TRY AND GET A VISUAL ON THE APCHING TFC BUT WAS LOOKING SOMEWHAT WESTWARD TOWARD EVENING ON A HAZY DAY AND COULD NOT ACQUIRE THE TFC. TCASII CHANGED THE ALERT FROM A TA TO AN RA WITH RED SQUARE INDICATING APCHING TFC WAS 200 FT BELOW OUR ALT. TCASII COMMANDED A CLB AT A RATE OF OVER 3000 FPM. I CLBED AT 3200 FPM AND NOTICED AN ACFT THAT APPEARED TO BE A SWEPT-WING BUSINESS TYPE JET; CPR Y; PASS UNDER MY ACFT. AFTER TCASII RA ENDED; THE COPLT CALLED CHICAGO APCH TO RPT THE TFC AND ADVISE APCH CTL OF OUR CLB. WE HAD CLBED APPROX 1000 FT. APCH CTL ACKNOWLEDGED THAT OUR PRESENT ALT WAS '1000 FT ABOVE THE TFC.' WE STATED WE WERE RETURNING TO OUR ASSIGNED ALT OF 2700 FT MSL. WE INTERCEPTED RWY 22R LOC AND GS. I REALLY BELIEVE IN TCASII. TCASII HELPED PREVENT US FROM GETTING ANY CLOSER TO THE TFC BEFORE EVASIVE ACTION WOULD HAVE BEEN TAKEN. WE MIGHT HAVE MISSED EXTREMELY CLOSE WITH THE TFC HAD IT NOT BEEN FOR TCASII. SUPPLEMENTAL INFO FROM ACN 244546: I RPTED TO APCH CTL THAT WE WERE RESPONDING TO A TCASII RA AND WE WERE TOLD THAT THE TARGET WAS 1000 FT BELOW US; WHICH IT WAS AFTER COMPLETION OF THE RA FOLLOWING THE TCASII EVENT.