A CL65 FLT CREW ON THE PHX CHILY 1 SID DEP DSNDING TO FL270 EXPERIENCED AN NMAC WITH ACFT ON THE LAS FUZZY 6 STAR.

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AS CAPT ON AN ACR FLT FROM PHX AT FL280 ON THE CHILY 1 DEP FROM PHX; WE RECEIVED A TCAS RA. WE WERE 10-20 NM SE OF BTY VOR AND HAD JUST RECEIVED AN INSTRUCTION FROM ZLA TO DSND TO FL270. AS SOON AS I BEGAN THE DSCNT; WE RECEIVED THE TCAS RA INSTRUCTING US TO INITIATE A CLBING ESCAPE MANEUVER WHICH CONFLICTED WITH ATC'S INSTRUCTIONS. WE WERE NOT WARNED OF THE IMPENDING CONFLICT BY ATC. I DISENGAGED THE AUTOPLT; INCREASED THRUST AND CLBED FOLLOWING THE ESCAPE MANEUVER DEPICTED ON THE TCAS DISPLAY. WE CLBED 2000 FT TO FL300. WE NEVER HAD VISUAL CONFIRMATION WITH THE OTHER ACFT BUT IT APPEARED TO HAVE PASSED DIRECTLY BELOW US BY LESS THAN 500 FT (BASED ON THE INFO PROVIDED BY OUR TCAS). ZLA CLAIMED RESPONSIBILITY FOR THE CONFUSION WHICH HAD THE OTHER ACFT DSNDING THROUGH OUR ALT. SINCE WE WERE ON THE CHILY 1 DEP FROM PHX (ON THE 114 DEG RADIAL FROM BTY); AND THE OTHER AFT WAS ON THE FUZZY 6 ARR TO LAS VEGAS (ON THE 115 DEG RADIAL FROM BTY); IT APPEARS THAT THE OVERLAPPING CHILY 1 SID AND THE FUZZY 6 STAR NEAR BTY VOR CONTRIBUTED TO THE NEAR MISS.