2 B767 ACFT; APCHING RBS VOR FROM OPPOSITE DIRECTIONS WERE BOTH AT FL390. ACR Y RESPONDS IMMEDIATELY TO A TCASII RA COMMANDING A CLB. AT THIS TIME THE CTLR SEES THE PROB AND INSTRUCTS ACR X TO CLB AND TURN. ACR X; WHILE ASKING THE CTLR TO CONFIRM THE INSTRUCTION; ALSO RECEIVES A TCASII RA COMMANDING A CLB. BOTH ACFT SIMULTANEOUSLY OBSERVE EACH OTHER PASS.

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I WAS WORKING THE ELX SECTOR AT ZAU DURING MODERATE TFC WHEN FLOW CTL CAME UP BEHIND ME TO INFORM ME OF A NEWARK; NJ; IN-TRAIL PROB WITH 3 ACFT IN 3 DIFFERENT SECTORS ALL ABEAM OF EACH OTHER AND NEEDING 15 MI IN-TRAIL. WHEN I TURNED BACK TO THE SECTOR I HAD LOST MY TRAIN OF THOUGHT; FORGETTING ABOUT A FL390 PROB. I HAD BEGUN THE COORD NEEDED FOR THE NEWARK PROB WHEN THE FL390 PROB WAS BROUGHT TO MY ATTN. I TURNED ACR X TO A 340 DEG HDG AND CLBED TO FL410. I ATTEMPTED SEVERAL TIMES TO DSND AND TURN ACR Y BUT COULD NOT GET AN ANSWER. THE ACFT PASSED 2.7 MI AND 800 FT. SUPPLEMENTAL INFO FROM ACN 407385: WE (ACR Y) RECEIVED AN RA WITH A CLB COMMAND. WE IMMEDIATELY STARTED CLB; WE SAW THE TFC ON TCASII AT WHICH TIME THE CTLR COMMANDED A R TURN WITH A DSCNT. WE STAYED WITH THE CLB KNOWING A CHANGE TO A DSCNT WOULD PUT US ON TOP OF THE TFC. WE VISUALLY SAW THE TFC PASS BY OUR L SIDE 1-3 MI OUT. SUPPLEMENTAL INFO FROM ACN 407125: WE WERE IN LEVEL CRUISE AT FL390; PROCEEDING DIRECT TO THE ROBERTS VOR. ZAU SAID 'ACR X; TURN R TO 340 DEGS; CLB TO FL410.' SINCE THIS WAS APPROX A 100 DEG HDG CHANGE; AND WE WERE ALREADY AT OR NEAR OUR MAX ALT FOR OUR GROSS WT; I ASKED THE CTLR TO 'CONFIRM' THAT THE CLRNC WAS FOR ACR X. HE DID; SO I (ACR X) DIRECTED THE FO TO TURN AND CLB. I THEN HEARD ACR Y HVY RPT THAT THEY WERE RESPONDING TO AN RA. OUR TCASII THEN GAVE US AN RA TO 'CLB.' OUR RELIEF PLT THEN POINTED OUT THE TFC AT 11 O'CLOCK POS.