A300 AND B737 HAD A COORDINATED TCASII RA IN ZDC CLASS A.

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ACR X; LEVEL FL270; MIA TO JFK. INTRUDER ACFT ACR Y WAS CLBING THROUGH OUR ALT. INTRUDER RELATIVE ALT AT TA WAS FL260 AND CLBING TO 11:30 O'CLOCK POS. INTRUDER RELATIVE ALT AT RA WAS FL265 12 O'CLOCK POS. MONITOR VERT SPD 1200 FPM CLB ESTABLISHED. CALLED ATC. NO WARNING OR CALL FROM ATC PRIOR. ATC RESPONDED WITH A 30 DEG R TURN (IMMEDIATE) FOR ACR X ALSO TO ACR Y. WE CLBED AT 600 FT FL276 BEFORE CLR OF CONFLICT; THEN DSNDED BACK TO ASSIGNED ALT. ATC PUT US IN A SIT THAT WAS DANGEROUS. WE WERE NOT TOLD OF AN ACFT CLBING THROUGH OUR ALT OPPOSITE DIRECTION CONVERGING. TCASII PREVENTED THE SIT FROM BEING WORSE THAN IT WAS. SUPPLEMENTAL INFO FROM ACN 465126: CRUISING AT INTERMEDIATE ALT OF FL260. WE WERE CLRED TO CLB AND MAINTAIN FL280. FO (I) BEGAN CLB IN VERT SPD AT 1000 FPM. AT FL264 WE GOT A TA. CONFLICTING ACFT WAS AT ABOUT 11-12 O'CLOCK POS AND 15 MI. WE CONTACTED ATC (ZDC) TO QUESTION. I DECREASED RATE OF CLB. ATC ANSWERED WITH INSTRUCTIONS TO TURN 30 DEGS R AND INCREASED RATE OF CLB. SIMULTANEOUSLY WE RECEIVED AN RA TO DSND. WE BEGAN A DSCNT AS WE TURNED. I TURNED OFF AUTOPLT TO MAKE THE TURN AND DSCNT TO COMPLY WITH THE RA. CAPT INFORMED ATC THAT WE WERE DSNDING AND TURNING TO COMPLY WITH THE RA. ATC GAVE US CLB TO FL260. OTHER ACFT RPTED RESPONDING TO AN RA TO CLB. THE CAPT HAD OTHER ACFT (ACR X) IN VIEW AT ALL TIMES. AFTER WE PASSED THE ACFT WE WERE ONCE AGAIN CLRED TO CLB AND MAINTAIN FL280.