B737-200 CREW AND AN UNKNOWN ACFT HAD A TCASII RA IN ZDC CLASS A AIRSPACE.

Narrative

WHILE CLBING OUT WITH ZDC; WE RECEIVED A TCASII ALERT THAT INSTRUCTED US TO CLB INTO TFC AND DEVIATE FROM ATC CLRNC. WE WERE CLBING AT 500-700 FPM AND WE WERE AT FL257 CLBING TO FL260 WHEN WE GOT A TFC CAUTION LIGHT AND AURAL WARNING. THIS WAS QUICKLY FOLLOWED BY AN ALERT WITH 'CLB; CONTINUE CLB' AURAL WARNING. THE FO WAS FLYING. HE DISCONNECTED THE AUTOPLT AND PULLED THE NOSE UP TO FOLLOW THE VVI WHICH WAS INDICATING A SMALL GREEN ARC AT ABOUT 1000 FPM CLB. I MADE A CALL GO ZDC TO ADVISE THAT WE WERE CLBING THROUGH OUR ASSIGNED ALT FOR A TCASII ALERT. AT ABOUT THIS SAME TIME; I VISUALLY ACQUIRED THE TFC AT MY L 9:30N O'CLOCK POS; SLIGHTLY HIGH AND ABOUT 4 MI AWAY ON A COURSE TO PASS BEHIND US BY ABOUT 2 NM. I TOLD THE FO THAT WE SHOULD DSND; SINCE THE TFC WAS NOT A FACTOR AND WAS ABOVE US. WHILE I WAS TALKING; THE TCASII CHANGED TO 'DSND' AURAL ALERT. WE DSNDED BACK TO FL260 AND ZDC INFORMED US THAT THE CONFLICTING TFC WAS LEVEL AT FL270. WE WROTE UP THE TCASII UPON ARR SO MAINT COULD CHK OUT ITS OP.