CRITICAL CONFLICT DURING A VISUAL SEPARATION CLB BTWN A CLBING LR25 AND A CLBING MD80 20 MI NW OF BNA; TN.

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AS WE CLBED OUT OF JWN NEAR BNA; WE WERE TOLD TO PROCEED ON COURSE; BUT MAINTAIN SEPARATION FROM AN MD80 AT OUR 2 O'CLOCK POS. THE CAPT PICKED A HEADING THAT WOULD PASS BEHIND THE MD80; AND WE DECIDED TO OUT-CLB THE ACFT. AS WE KEPT OUR EYE ON THE MD80; IT CAME APPARENT WE BOTH PICKED A DIRECTION THAT WAS CONFLICTING WITH THE OTHER ACFT OR THE MD80; TURNED MORE IN OUR DIRECTION. WE DID NOT HAVE PAX AND HAD TO TURN 30 DEGS R AND LEVEL OFF TO AVOID CONFLICT. THE MD80 RECEIVED AN RA. THE PROB COULD HAVE BEEN THE ANGLE WE WERE APCHING THE ACFT WAS DECEIVING. IT GAVE THE IMPRESSION SEPARATION WAS OK. SUPPLEMENTAL INFO FROM ACN 597725: I WAS CLBING OFF OF JWN AND THE MD80 WAS CLBING OFF OF BNA. WE APPEARED TO BE CLBING ON PARALLEL COURSES. I WAS INCREASING MY RATE OF CLB TO OUT CLB THE MD80. IT WAS GOING OK WHEN IT APPEARED THE MD80 TURNED TOWARDS ME. REEVALUATING THE SIT; I ELECTED TO TURN R WITH A 30 DEG BANK TURN. WHEN I WAS ABOUT 90 DEGS FROM HIS COURSE; THE MD80 RPTED AN RA AND SLOWED HIS RATE OF CLB. I BELIEVE THE RA WAS FROM THE PROJECTED COURSES AND NOT THE ACTUAL COURSE RESULTING FROM MY COURSE CHANGE. I DON'T BELIEVE THIS WAS A NEAR MISS AFTER TALKING TO THE CTLR FROM NASHVILLE APCH. THEY MADE THE STATEMENT THAT IT WAS A 'CLEAN' OP AND THAT VISUAL SEPARATION MINIMUMS WERE MAINTAINED.