B757 FLT CREW BEING VECTORED BY MIA TRACON EXPERIENCED TCASII RA AT 8000 FT WITH B737 WHO FAILED TO STOP CLB AT 7000 FT.

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WE WERE LEVEL AT 8000 FT; HEADING EBOUND AND ABOUT 20 MI W OF MIA. THE AUTOPLT WAS ENGAGED. MIA APCH WAS PLANNING ON PUTTING US ON THE DOWNWIND FOR LNDG ON RWY 30. THERE WAS 1 LONE CUMULOUS CLOUD BTWN US AND THE ARPT. THE TOP OF THIS CLOUD WAS SLIGHTLY BELOW US AT ABOUT 7800 FT. I OBSERVED THE APCHING B737 AHEAD AND TO MY L WHEN IT WAS AT AROUND 5000 FT CLBING TO 7000 FT (I THOUGHT). AT THAT TIME HE APPEARED TO BE TURNING SW TO S. AT THIS POINT THE B737 FLEW INTO THE SINGLE CUMULOUS CLOUD (OR MAYBE BEHIND IT) AND WE WERE ALMOST ON TOP OF THIS CLOUD. THE TCASII SOUNDED AND I DISCONNECTED THE AUTOPLT AND STARTED A L CLBING TURN AS THE B737 BROKE OUT FROM THE CLOUD (OR FROM BEHIND IT) IN FRONT OF US AND SLIGHTLY SLOWER BUT CLBING. IT HAPPENED VERY QUICKLY AND IT WAS OVER IN A MOMENT. I CAN'T TELL YOU HOW CLOSE WE CAME TO THE B737 OR IF HE EXCEEDED 8000 FT. DURING THE EVENT; THE FO COMMUNICATED WITH MIA AND WE LANDED WITHOUT ANY FURTHER PROBS. AFTER CLRING CUSTOMS AND GETTING THE PAPERWORK FOR THE NEXT LEG; I DID CALL THE ATC FACILITY AND CHATTED FOR A FEW MINS WITH THE GENTLEMAN WHO WAS AT THAT MOMENT WRITING THE ATC RPT ON THIS INCIDENT. THIS IS MY BEST RECOLLECTION OF THE EVENT.