TURBOJET ACR INITIATE EVASIVE MANEUVER DUE TO TCASII OBSERVED TARGET IN VICINITY OF ODI VORTAC UNDER ZMP CTL CLBING THROUGH 5000 FT.

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CLBING OUT OF LSE WE WERE CLRED DIRECT ODI AND TOLD TO CONTACT ZMP. WE CHKED IN WITH ZMP THROUGH 3000 FT; CLBING TO 5000 FT DIRECT ODI. WE WERE CONFIRMED DIRECT ODI AND CLRED TO 10000 FT. ABOUT 1 MI E OF THE ODI VOR; TFC POPPED UP ON OUR TCASII SHOWING OUR ALT; CLBING; ABOUT 1/2 - 1 MI AHEAD TO OUR L. THERE WAS HAZE AND IT WAS HARD TO SEE W INTO THE SUN. THE FO WAS FLYING AND I TOLD HIM TO 'CLB HARD' AS THERE WAS NO RA FROM TCASII AND JUST THEN A CESSNA CALLED MSP AT 4500 FT OVER THE ODI VOR. ON THE TCASII THE TARGET MERGED WITH OUR ACFT WHILE 200 FT BELOW US. WE NEVER SAW THE TFC AND ZMP SAID THEY NEVER SAW IT IN TIME. I DON'T KNOW WHY WE GOT NO RA FROM THE TCASII. PERHAPS WE WERE RIGHT ON THE THRESHOLD OF AN RA AND THE TFC WAS NOT DISPLAYED VERY LONG. CONTRIBUTING FACTORS WERE THE HAZE AND THE CONDENSING OF TFC NEAR VORS. I BELIEVE IF WE HAD NOT BEEN ALERTED TO THE TFC BY THE TCASII AND NOT INCREASED OUR RATE OF CLB WE WOULD HAVE COME A LOT CLOSER TO THE ACFT. AS WE WERE AVOIDING THIS TFC; WE CLBED THROUGH A BROKEN LAYER OF CLOUDS. I DON'T REMEMBER THE EXACT ALTS INVOLVED; BUT ONE OF MY FIRST THOUGHTS WAS THAT THIS VFR TARGET WAS NOT 500 FT BELOW THE CLOUDS. PERHAPS THE CESSNA PLT HAD ALLOWED THE BUMPY AIR TO PUSH HIM UP HIGHER THAN HE SHOULD HAVE AND ALLOWED HIS CLB RATE TO NEAR OURS.