ACR BE1900 ACCEPTS VISUAL SEPARATION CLRNC FROM PREVIOUSLY DEPARTING E120. AS ATC TURNS BOTH ACFT SBOUND; THE BE1900 CONTINUES VISUAL SEPARATION FROM E120. BE1900 FLC INITIATES TCASI MANEUVER WHEN CLE TRACON TURNS E120 INTO BE1900.

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AFTER DEPARTING RWY 5R AT CLEVELAND HOPKINS ARPT IN CLEVELAND; OH; WE (X) WERE TOLD TO MAINTAIN VISUAL SEPARATION WITH AN EMB120 (Y) WHICH HAD JUST DEPARTED AHEAD OF US. BOTH OUR ACFT AND THE EMB120 WERE ASSIGNED A 095 DEG HDG AFTER DEP; AND THE SEPARATION DURING THE CLBOUT WAS ONLY ABOUT 1/2 MI. AFTER PASSING ABOUT 2000 FT MSL; WE WERE ASSIGNED A R TURN TO A 180 DEG HDG; AND SHORTLY AFTER THE EMB120 WAS ALSO ASSIGNED THE SAME R TURN TO THE 180 DEG HDG. BOTH ACFT WERE ONLY ABOUT 1/2 MI HORIZ SEPARATION ON A S HEADING. INITIALLY THE EMB120 WAS ABOUT 700 FT ABOVE US DURING THE VECTORING S. AFTER BOTH ACFT WERE ESTABLISHED ON THE 180 DEG HDG; WE BEGAN TO OUT CLB THE EMB120. AT APPROX 5500 FT MSL; WE WERE AT THE SAME ALT AS THE EMB120; BOTH ACFT STILL CLBING TO HIGHER ASSIGNED ALTS AND BOTH STILL ONLY 1/2 MI APART. THE EMB120 BEING TO OUR L. AT THIS TIME; CLEVELAND DEP CTL DIRECTED THE EMB120 TO TURN R TO A HDG OF 260 DEGS. UPON TURNING TO THIS HEADING; THE EMB120 WAS HEADING DIRECTLY TOWARDS US AT THE SAME ALT. AFTER SEEING THIS OCCUR; AND AFTER HAVING RECEIVED A TFC ALERT (TA) ON OUR TCASI; WE INITIATED A STEEP CLB (ABOUT 3000 FPM) UP TO AVOID THE ON-COMING EMB120. AT THIS TIME; THE EMB120 PLTS ALSO RECEIVED A TA AND SAW US VISUALLY AND BEGAN A DSCNT TO AVOID US. OUR BE1900 AND THE EMB120 MISSED EACH OTHER BY APPROX 600 FT VERTLY. NO DAMAGE OR INJURIES OCCURRED ON EITHER ACFT. THE REMAINDER OF THE FLT OCCURRED WITHOUT INCIDENT.