FLC OF LGT ACR ACFT TOOK EVASIVE ACTION AND CLBED ABOVE ASSIGNED IN RESPONSE TO A TCASII RA.

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ATC RPTED UNCTLED TFC AT 12 O'CLOCK AT 10800 FT. TCASII SOUNDED 'TFC; TFC.' WE LOOKED AT TCASII SCREEN AND SAW TARGET ABOUT TO MERGE WITH OUR ACFT ON 10 MI SCALE AND RECEIVED TCASII 'CLB; CLB.' ROLLED L AND THROTTLE PUNCHED FORWARD BY CAPT AND CLBED. HDG CHANGED 30 DEGS AND LEVELED AT 11700 FT. FO SAW MLG TYPE ACFT PASS TO L SIDE OF ACFT AND WE STARTED DSCNT BACK TO ASSIGNED ALT. ATC CONCERNED WITH OUR ACTION TO AVOID TFC. NO INJURIES. HAVING TCASII DISPLAY ON RADAR SCREEN WAS KEY IN DECISION TO CLB AS IT RECONFIRMED TCASII ALERT. TCASII WARNING OF 'TFC' WAS WHAT GOT US LOOKING OUTSIDE AND AT TCASII DISPLAY. SMOOTH AND PROMPT AVOID MANEUVER KEPT US FROM HAVING INJURIES OR ACFT DAMAGE. WITHIN 2 MINS ACFT FOLLOWING US ON NANCI-4 ARR TO LGA ALSO ENCOUNTERED THIS ACFT AND DID EVASIVE MANEUVER TO AVOID. WE DID NOT HEAR OR WERE NOT TOLD WHY THIS ACFT WAS IN OUR AIRSPACE OR WHERE HE CAME FROM. IF HE WAS IN LEVEL FLT WE ARE LUCKY THAT HE MUST HAVE BEEN USING DIFFERENT ALTIMETER SETTING OR FLYING OFF ALT IF HE DID INTEND TO FLY AT 11000.