2 ACR ACFT MISS EACH OTHER BY AN UNKNOWN DISTANCE; ALLEGED TO BE 1 AND ONE HALF MI HORIZLY AND 300 FT VERTLY BY THE FLC. ATC RECORDS NO LOSS OF SEPARATION. XING TFC; INITIALLY AT 17000 FT. ATC HAD DSNDED ACR Y TO 16000 FT JUST AS THE TCASII RA'S HAD MADE A RESOLUTION DETERMINATION FOR BOTH ACFT.

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AFTER DEP FROM JFK; WE WERE LEVEL AT 17000 FT ON RADAR VECTORS FROM DEP CTL. WE WERE ON A HDG OF 330 DEG ASSIGNED; WHEN CLRED DIRECT TO SPARTA VOR (SAX). WHILE IN THE L TURN TO SAX; A TARGET APPEARED ON TCASII SHOWING LEVEL AT 17000 FT; APPROX 8 NM AHEAD. I DID NOT NOTICE THE DIRECTION OF TRAVEL; AS I WAS LOOKING OUTSIDE TO TRY TO MAKE VISUAL CONTACT. AT THIS TIME; WE WERE IN A HAZE LAYER THAT RESTR VISIBILITY TO APPROX 2 NM. I THEN HEARD THE TCASII RA TO 'DSND; DSND NOW.' THE CAPT DISCONNECTED THE AUTOPLT AND PITCHED THE ACFT TO THE DSCNT TARGET VALUE INDICATED. AT THE SAME TIME; I HEARD ATC INSTRUCT THE OTHER ACFT TO DSND TO 16000 FT. THEY REPLIED THAT THEY HAD A TCASII RA TO CLB. AS WE CONTINUED TO DSND; THE OTHER ACFT GOT CLOSER AND I NOTICED THAT IT WAS SHOWING 300-500 FT BELOW US AND DSNDING. AT THIS POINT THE CAPT LEVELED OUR ACFT AT 15000 FT AND THE OTHER ACFT PASSED UNDER US AND LEFT OUR 4 O'CLOCK POS. THE TCASII THEN ANNOUNCED 'CLR OF CONFLICT.' WE THEN CLBED BACK TO 17000 FT. I BELIEVE WHAT CONTRIBUTED TO THIS INCIDENT FURTHER WAS THE OTHER ACFT FOLLOWING ATC'S INSTRUCTIONS TO DSND; INSTEAD OF CLBING AS TCASII INSTRUCTED AND WE ARE TRAINED TO RESPOND TO TCASII. THE ONLY VISUAL CONTACT WITH THE OTHER ACFT WAS WHEN IT WAS DEPARTING OUR AREA AT THE 4 O'CLOCK POS.