ACR X DSCNT THROUGH OCCUPIED ALT TCASII RA HAD LTSS FROM ACR Y. SYS ERROR.

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DURING DSCNT; CLRED TO FL190. CTLR INSTRUCTED ACR X TO LEVEL AT FL240. WE HAD ALREADY DSNDED TO ABOUT FL235. VERT SPD HAD BEEN APPROX 3500 FPM. NEARLY SIMULTANEOUSLY; TCASII TA OCCURRED; FOLLOWED VERY SHORTLY THEREAFTER BY AN RA OF 'CLB.' CLBED TO FL240 EXPEDITIOUSLY. CTLR ISSUED A TURN TO 180 DEGS (90 DEG TURN TO THE L). ACR Y PASSED 2 MI TO THE R ABOUT 1000 FT BELOW US. WOULD HAVE BEEN VERY CLOSE HAD EVASIVE ACTION NOT BEEN INITIATED. TCASII WORKED WELL AND WOULD HAVE RESOLVED CONFLICT. CTLR ALSO CAUGHT HIS ERROR IN A TIMELY MANNER.L THERE IS NO QUESTION THAT WE WERE CLRED TO FL190. A CONTRIBUTING FACTOR EFFECTING EVENTS: WE HAD BEEN LEVEL AT FL290 AND CLRED TO FL240 BY HIGH ALT CTLR. WHEN SWITCHED TO LOW ALT CTLR WE HAD NOT YET LEVELED AT FL240 AND WE WERE CLRED TO FL190. THERE WAS TSTM ACTIVITY IN THE TERMINAL AREA AT MIA SO THIS COULD HAVE BEEN DISTRACTING TO THE LOW ALT CTLR. SUPPLEMENTAL INFO FROM ACN 250240: ACR X TCASII TA WAS RECEIVED FOLLOWED BY AN RA TO CLB. ACFT WAS LEVELED AT FL240 FOLLOWED BY AN ATC VECTOR (L) TO A HDG OF 180 DEGS. OTHER ACFT WAS SIGHTED TO OUR R AS WE WERE IN THE L TURN. ACFT WAS APPROX 2 MI TO THE R AND BELOW OUR ACFT.