ALT OVERSHOOT IN CLB FOR ACR FLT RESULTS IN LOSS OF SEPARATION AND A TCASII RESPONSE.

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WE WERE GIVEN FL280 TO LEVELOFF. WITHIN 1000 FT OF LEVELOFF I CHKED A FREQ ON THE CHART TO MAKE SURE I WAS HEADED IN THE RIGHT DIRECTION. THE SUN WAS EXCEPTIONALLY BRIGHT AND BLINDING ME SLIGHTLY. THIS CAUSED ME TO TAKE A LITTLE LONGER FINDING THE INFO. I WAS THROUGH CHKING FOR THE FREQ AT THE SAME TIME WE HEARD 'TFC' FROM THE TCASII. I LOOKED AT THE TCASII AND ALTIMETER AND SAW PLANE (Y) AT THE SAME TIME AS WE WERE JUST THEN GOING THROUGH FL280. THE OTHER PLANE (Y) WAS COMING FROM L TO R AT FL290. I ROLLED THE PITCH COMMAND ON THE AUTOPLT TO THE DSCNT RANGE IMMEDIATELY. THE CAPT DID NOT REALIZE I WAS USING THE AUTOPLT TO DSND. AT ABOUT 28250 FT WE HEARD 'MONITOR VERT SPD' FROM THE TCASII. THE CAPT SAID TO GET THE AUTOPLT OFF THEN HE GRABBED THE YOKE AND KICKED OFF THE AUTOPLT AND PUSHED HARD ON THE YOKE. WHEN THE AUTOPLT KICKED OFF IT CAUSED A SLIGHT PITCH UP AGAIN. I WAS ALSO HOLDING THE YOKE AT THIS TIME. AT THIS TIME WE ALSO HEARD 'DSND; DSND' FROM THE TCASII. AT THIS POINT I COULD VISUALLY SEE THAT WE WERE IN NO DANGER OF COMING ANY CLOSER TO THE OTHER ACFT (Y); OUR MAX ALT WAS 28300 FT; AND THE OTHER ACFT (Y) WAS ALMOST DIRECTLY OVERHEAD. I RETURNED OUR ACFT (X) TO FL280 AND CONTINUED THE FLT. THE APCHING ACFT AT FL290 WAS NOT POINTED OUT TO US BY ATC.