B767 REACTED TO A TCASII RA AND SIMILAR CALL SIGN CONFUSION WHILE UNDER ZAU CTL.

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NEAR BDF VOR UNDER ZAU CTL WE WERE PLACED IN A CONFLICT SIT REQUIRING MANEUVERING IN RESPONSE TO AN RA. CTR TOLD US TO CLB 2000 FT WHEN HE REALLY MEANT ACFT Y TO CLB. THERE HAD BEEN A FREQ MIX-UP WITH ACFT Y WHILE WE WERE ON ZKC SO WE WERE SAYING ACFT X BEFORE AND AFTER OUR NUMBER CALL FOR CLARITY. THE CTLR STATED THAT HE HAD JUST TOLD THE WRONG ACFT X TO CLB. AFTER CLBING 1000 FT THE TFC ALERT SOUNDED AND I OBSERVED THE ACFT 10 MI AHEAD 1000 FT HIGHER AND APPEARING TO BE HEADED DIRECTLY TOWARD US ON THE TCASII SCREEN. I CLICKED THE AUTOPLT OFF AND STARTED TO LEVEL WHEN THE RA TRIGGERED A DSCNT OF 1000 FPM NEARLY SIMULTANEOUSLY WITH A CALL FROM THE CTLR TO DSND. I FOLLOWED THE RA TO 1000 FPM DOWN AND THEN TO HOLDING LEVEL UNTIL AFTER PASSING BELOW THE TFC. I DOUBT THAT WE EVER GOT CLOSER THAN 900 FT VERTLY AS BOTH OF US WERE RESPONDING TO THE RA. THE DSCNT WAS SMOOTH TO THE POINT THAT THE RELIEF PLT IN THE CABIN WAS NOT AWARE THAT IT HAD OCCURRED. THE RA; ACFT; AND CTLR ALL REACTED CORRECTLY WHEN THE CONFLICT BECAME OBVIOUS.