AN SF34 FLC EXPERIENCED AN NMAC WITH AN MU2 CLBING THROUGH THEIR ALT. THE FLC RESPONDED TO THE TCASII RA; BUT THEN CONFLICTED WITH TFC ABOVE. NMAC. EVASIVE ACTION CLB.

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IN CRUISE FLT AT 16000 FT; INBOUND ON THE 046 DEG RADIAL TO MIDDY INTXN; RECEIVED ATC ADVISORY OF TFC; AN MU2; 12 O'CLOCK; 3 MI AND CLBING. AT THE SAME TIME; RECEIVED A TCASII TA. AUTOPLT WAS ENGAGED AND BOTH PLTS BEGAN VISUAL SCAN FOR TFC. APPROX 5 OR 6 SECONDS AFTER TA; RECEIVED TCASII RA 'CLB; CLB; CLB.' CONFLICT RESOLUTION INDICATED CLB OF 1500 FPM TO 2000 FPM. PF DISENGAGED AUTOPLT WITH PITCH TRIM AND INITIATED CLB WITHIN ABOUT 3 SECONDS. AT A VERT SPD OF APPROX 1000 FPM; TCASII ADVISED 'MONITOR VERT SPD.' PF THEN MADE VISUAL CONTACT WITH MU2 AS IT PASSED UNDER THE NOSE AND TCASII INDICATED THE ACFT TO BE 300 FT BELOW AND CLBING. ATC THEN CONTACTED US TO INQUIRE ABOUT OUR ALT; WE WERE APPROX 700-800 FT ABOVE ASSIGNED ALT WHEN WE BEGAN OUR DSCNT BACK TO 16000 FT. PNF INFORMED ATC THAT WE HAD RESPONDED TO A TCASII RA. WE WERE THEN CLRED TO A LOWER ALT AND SEVERAL MINS LATER INSTRUCTED TO CONTACT ATC BY TELEPHONE AFTER LNDG. DURING THE TELEPHONE CONVERSATION WE WERE INFORMED THAT OUR CLB IN RESPONSE TO THE TCASII RA HAD CONFLICTED WITH TFC AT 17000 FT; HOWEVER; WE DID NOT RECEIVE A TA FOR THE 17000 FT TFC FROM EITHER ATC OR TCASII. I BELIEVE THIS SIT WAS CAUSED BY ATC FORGETTING TO DSND US FOR THE ARR INTO MEMPHIS. FROM MY POINT OF VIEW I DID THE ONLY THING THAT I COULD HAVE DONE; I COMPLIED WITH THE CONFLICT RESOLUTION INDICATED BY THE TCASII AND AS SOON AS PRACTICAL; NOTIFIED ATC OF THOSE ACTIONS.