ALTDEV ALT EXCURSION AFTER TCASII RA RECEIVED ON UNKNOWN VFR TFC IN AREA.

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JUST AFTER LEVEL OFF AT 1400 FT ON FLT FROM TUS TO PHX; HAD YELLOW TCASII ALERT LIGHT. TFC INDICATED 400 FT BELOW US; LEVEL; 12 O'CLOCK. I COULD NOT QUERY ATC BECAUSE ATC WAS TALKING WITH ANOTHER ACFT. PNF AND SO COULD NOT SEE TFC (WE NEVER DID). I ANNOUNCED MY INTENTION TO CLB IF TCASII COMMANDED. WITHIN APPROX 10 SECONDS WE HAD A RED TCASII WARNING WITH A 2000 FPM CLB COMMAND. I IMMEDIATELY BEGAN TO CLB. JUST AFTER STARTING CLB WE WERE ABLE TO TALK TO ZAB ARTCC AND TOLD THEM WE WERE CLBING TO ABOUT 15000 FT TO AVOID TFC AND THAT WE WISHED TO DSND BACK TO 14000 FT AS SOON AS WE WERE PAST THE TFC. ARTCC AGREED. MY MAX ALT WAS 15200 FT; TOTAL TIME OFF ATL WAS ABOUT 2-3 MINS. WHEN ATC HAD TIME; THEY QUERIED US. WE EXPLAINED WE HAD A RED TCASII AVOIDANCE COMMAND. THE TCASII WORKED VERY WELL. MIXING VFR TFC AND IFR JET TFC IS NOT SAFE. USUALLY ATC TELLS US ABOUT POTENTIAL TFC CONFLICTS BEFORE TCASII SEES THEM. WHY DIDN'T ZAB SEE THE POTENTIAL CONFLICT? WITHOUT THE TCASII THIS WOULD HAVE BEEN A NEAR MISS OR WORSE. HAVING THE YELLOW TCASII ALERT FOR A FEW SECONDS ALLOWED US TIME TO EVALUATE THE THREAT AND TO PLAN THE MANEUVER. ANY DELAY IN INITIATING THE CLB WOULD HAVE BEEN VERY DANGEROUS.