EMB145 EXPERIENCED TCASII TA-RA ON DSCNT AT FL220 IN ZOB AIRSPACE.

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FROM DAY TO LGA ON THE MIP3 ARR INTO LGA WE WERE GIVEN A DSCNT FROM FL280 TO FL180. AT APPROX 23.5 ATC TOLD US TO EXPEDITE THROUGH FL220; THEN TOLD US TO TURN OFF THE ARR TO HEADING 090 DEGS. AS WE FOLLOWED THE INSTRUCTIONS WE RECEIVED A TCASII TA; 'TFC; TFC.' THE FO TURNED OFF THE AUTOPLT; TIGHTENED THE TURN AND INCREASED THE RATE OF DSCNT. ATC THEN ADVISED US TO INCREASE THE RATE OF TURN. I TOLD THE CTLR WE ALREADY HAD AND THAT OUR TCASII HAD GONE OFF. WITH 35 DEG L BANK AND APPROX 3500-4500 FPM DOWN ON THE VSI OUR PWR WAS IDLE AND AIRSPD PICKED UP FROM THE 280 KTS ASSIGNED EARLIER TO 320 KTS. WE RECEIVED A TCASII RA 'CLB; CLB' WITH THE LARGE RATE OF DSCNT AND THE GENERAL ATTITUDE OF THE ACFT WE ELECTED TO CONTINUE DOWN. THE RA WENT OFF AT APPROX FL227 TO FL225. TO REVERSE THIS COURSE OF ACTION WE FELT WOULD HAVE BEEN MORE HAZARDOUS THAN TO CONTINUE DOWN TO BELOW FL220 WHERE OUR TFC WAS. ONCE THE ACFT DSNDED BELOW FL210 THE TCASII ALERTED 'CLR OF CONFLICT' ATC THEN GAVE US DIRECT TO THE MAARC INTXN ON THE ARR AND WE CONTINUED DOWN TO FL180.