A CARJ HAS AN NMAC WITH A C172 AND THE FO INITIATES A CLB WHEN THE TCASII RA CALLED FOR A DSCNT.

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THIS INCIDENT TOOK PLACE DURING A REPOSITIONING FLT FROM PHX TO LAS. THE ONLY PAX ABOARD WAS A DEADHEADING COMPANY FLT ATTENDANT; SEATED IN THE FLT DECK JUMP SEAT. DURING CLBOUT FROM PHX; PHOENIX DEP ISSUED A CLB TO 9000 FT. JUST BEFORE WE LEVELED AT 9000 FT; DEP ISSUED TFC AT 1 O'CLOCK POS; 2 MI; 9000 FT; OPPOSITE DIRECTION. IMMEDIATELY AFTER THIS; TCASII GAVE US AN RA ON THE TFC. UPON SEEING THE RA; I DISCONNECTED THE AUTOPLT AND INITIATED A CLB. AFTER CLBING APPROX 200 FT; THE CAPT POINTED OUT THAT I HAD MISINTERPED THE RA COMMAND; I SHOULD HAVE BEEN DSNDING. I IMMEDIATELY REVERSED DIRECTION; BEGINNING A DSCNT. JUST AS I WAS BEGINNING TO DSND; THE CAPT AND I SPOTTED THE TFC. THE OTHER ACFT (A SMALL SINGLE-ENG; HIGH WING MONOPLANE) PASSED APPROX 50 FT ABOVE AND 500-1000 FT TO OUR R SIDE. AFTER WE RPTED THE TFC IN SIGHT; DEP ISSUED A FURTHER CLB. DEP QUERIED US AS TO WHETHER TCASII HAD COMMANDED A CLB; AND THE CAPT ADMITTED THAT IT HAD NOT. SUPPLEMENTAL INFO FROM ACN 630037: WE WERE TURNED N ON A HDG TO PASS 500 FT UNDER A C172. PHX DEP ADVISED US OF THE TFC. I WAS LOOKING OUTSIDE FOR THE TFC THAT WAS RPTED AT 12 O'CLOCK POS AND 7500 FT. WE WERE AT 7000 FT. AS EXPECTED; WE GOT A TFC ALERT. SINCE ATC HAD ADVISED US OF THE TFC; I DIDN'T EXPECT MY FO TO TAKE EVASIVE ACTION. WE HAD THE TFC ON TCASII. BEFORE I KNEW IT; HE STARTED A CLB AND ADDED PWR; EVEN THOUGH THE TCASII TOLD US TO DSND. SUPPLEMENTAL INFO FROM ACN 630284: JETLINER PASSED ME IN THE OPPOSITE DIRECTION AT ABOUT THE SAME ALT AND BTWN 1/2 AND 1 MI HORIZ SEPARATION W OF MY POS AND TURNING TO THE W. AFTER IT PASSED I WAS ADVISED OF TFC AT 1 O'CLOCK POS; TO WHICH I REPLIED 'IT JUST PASSED ME.' I WAS THEN TOLD 'TFC NO FACTOR.'