B737-800
Flight Crew Air Transport Pilot (ATP)
Situational Awareness
ATC Issue All Types; Conflict NMAC
Flight Crew FLC complied w / Automation / Advisory; Flight Crew Took Evasive Action
Procedure; Human Factors
Flying the SKIPS departure off of 26L we were given a turn direct to CRABI and climb from 7000 to 16000 after the initial southbound turn and climb to 7000 feet. As I began the left turn to the east and climbed out of 7000; we got a TA followed quickly by an RA. The command was a significant descent. I was still hand flying; so I complied with the RA expeditiously. I used the turn to keep from negative G'ing our jet as I transitioned from a climb to a rapid descent. The First Officer said it looked like a TBM or similar single engine mono plane; who was also in a rapid descent. As we passed the intruder aircraft; the First Officer said 'That was close'. He told ATC we were complying with an RA. He thought we missed by 200-300 feet laterally and passed at close to the same altitude. After the maneuver we resumed our climb and turn to CRABI; notifying ATC. We continued to our destination. The passengers and Flight Attendants were still strapped in; so it was just a close call. No damage or injuries.ATC steered us into the path of another aircraft. We were just NW of Tamiami airport. Probably a lot of VFR traffic on a day like this. I think in this case the question is better answered by ATC or the intruder aircraft? What I would like to know is why the other aircraft was also in a steep descent. Was he not complying with his TCAS or not given an RA?  Seems unusual for TCAS to command descents to both aircraft.
B737 Captain reported an NMAC with a single engine aircraft shortly after departing MIA.