AN MLG CREW RESPONDED TO A TCASII COMMAND TO 'CLB.' THEY DID; AND MISSED AN SMA BY ABOUT 300 FT.

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AT THE TIME OF THE NMAC; WE WERE ON A RADAR VECTOR WITH CLRNC TO DSND TO 11000 FT. AT ABOUT 120000 AT TCASII TA SOUNDED WHICH RAPIDLY WARNED 'REDUCE DSCNT.' I OBSERVED ANOTHER ACFT WELL BELOW US ON A CONVERGING COURSE. I THOUGHT THIS WAS THE TCASII TARGET. THE FO WAS TALKING TO APCH CTL WHICH CONFIRMED WE WERE CLRED TO 11000. TCASII THEN WARNED 'CLB.' A CLB WAS INITIATED IMMEDIATELY. WITHIN 5 SECONDS AND ABOUT 100-200 FT INCREASE IN ALT; A SINGLE ENG LOW WING BLUE AND WHITE ACFT PASSED DIRECTLY BELOW US IN THE OPPOSITE DIRECTION. I THINK HE WAS ABOUT 300 FT BELOW US. WE WERE NEVER ADVISED OF THIS ACFT. TCASII PROBABLY PREVENTED A MIDAIR. WE ADVISED APCH CTL OF THE CLB AND EVASIVE ACTION; AND HE CONFIRMED WE WERE CLR OF IT; AND GAVE US A TURN TO AVOID THE CONVERGING ACFT. APCH CTL WAS VERY BUSY AT THE TIME. SUPPLEMENTAL INFO FROM ACN 215214: PF OBSERVED A REGIONAL CARRIER TURBOPROP ACFT TO OUR L AND BELOW ON A CONVERGING COURSE; AND ASSUMED HE HAD TFC IN SIGHT. FLT ENGINEER RELAYED THE CLB COMMAND TO PF WHO THEN ADDED PWR AND CLBED. PF THEN OBSERVED A SINGLE ENG LOW WING GA ACFT PASS DIRECTLY BELOW US ABOUT 200-400 FT VERT SEPARATION 180 DEG OPPOSITE COURSE. NO TFC CALL FROM APCH CTL. IN FACT; APCH THEN GAVE A CALL TO AN INCORRECT CALL SIGN FOR US TO TURN R ABOUT 60 DEGS. BY NOW THE REGIONAL CARRIER TURBOPROP ACFT WAS STILL CONVERGING; AND WE WERE BECOMING CONCERNED ABOUT IT. APCH CTL THEN ISSUED A TURN TO US USING THE PROPER CALL SIGN AND ALSO TURNED THE REGIONAL CARRIER TURBOPROP ACFT AWAY FROM US.