ALTDEV ALT EXCURSION IN FLC RESPONSE TO TCASII RA.

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OUR FLT WAS GIVEN A CLRNC TO CROSS ROBERT INTXN AT 9000 FT. APPROX 5 MI NE OF ROBERT (STILL DSNDING TO CROSS ROBERT AT 9000 FT) WE RECEIVED A TA FOLLOWED ALMOST IMMEDIATELY BY AN RA 'CLB XING CLB.' THE COPLT (WHO WAS THE PF) IMMEDIATELY COMPLIED AND I NOTIFIED ATC THAT WE WERE CLBING ON A TCASII ALERT. AT THIS TIME I NOTED OUR ALT WAS 9800 FT AND CLBING. ATC RECOMMENDED A DSCNT; BUT AS PER COMPANY AND FAA POLICY WE CONTINUED THE CLB. WE HELD THE CLB AS PER THE TCASII AND GOT A 'CLR OF TFC' AT APPROX 10800 FT. AT THIS TIME I GOT A VISUAL ON A WDB AT 11 O'CLOCK; APPROX 1 MI AND BELOW US. WE LEVELED AT 11000 FT AND ASKED ATC FOR INSTRUCTIONS. HE GAVE THE WDB A CLRNC TO MAINTAIN 9000 FT AND GAVE US A CLRNC TO MAINTAIN 10000 FT. I AM WRITING TO RPT TO EMPHASIZE THE IMPORTANCE OF COMPLIANCE WITH TCASII. OUR CLRNC WAS TO CROSS ROBERT AT 9000 FT AND WE WERE COMPLYING WITH IT. I SUSPECT THE WDB'S CLRNC WAS TO CROSS ROBERT AT 10000 FT. IN THIS EVENT; THE CTLR WOULD BELIEVE IT WOULD HAVE BEEN BETTER FOR US TO CONTINUE OUR DSCNT TO 9000 FT. HOWEVER; AS TCASII CROSSTALKS AND CONSIDERS CLOSURE RATE; ITS CLB SIGNAL TO US AND 'DSND' SIGNAL TO THE WDB WOULD BE UNDERSTANDABLE. BY COMPLYING WITH THE TCASII WE AVOIDED THE OTHER ACFT. HAD WE LEVELED AND THEN DSNDED AS ADVISED BY THE CTLR; I HAVE NO DOUBT THAT THE ACFT WOULD HAVE CAME MUCH CLOSER AS OUR CLB WOULD HAVE STOPPED AND OUR DSCNT BEGAN AT THE WDB'S ALT.