CL600 FLT CREW RESPONDED TO TCAS RESOLUTION ADVISORY DURING CRUISE. FLT CREW BELIEVES THE INTRUDING ACFT WAS ALSO RESPONDING TO AN RA.

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WHILE LEVEL AT FL290 APCHING FLABB INTXN ON ARR TO CLT (JOHNS1) WE RECEIVED A TCAS TA; THEN AN RA COMMANDING US TO CLB AT 1000 FPM. I IMMEDIATELY COMPLIED WITH THE RA. THE COMMAND CONTINUED AT 1000 FPM UNTIL REACHING FL300. WE ADVISED ATC. WE RECEIVED NO TA FROM ZTL. I THINK THE B737 BELOW US WAS RESPONDING TO HIS OWN RA AND CLBED TO OUR ALT. THE B737 CLBED UP TO OUR ASSIGNED ALT; POSSIBLY THROUGH IT; AND PASSED DIRECTLY BELOW US AS SEEN BY FO AND AS SHOWN ON OUR MFD. I FIRMLY BELIEVE; BOTH IMMEDIATELY AFTER THE EVENT; AND NOW AFTER REFLECTION; THAT THE TCAS UNIT IN OUR ACFT SAVED THE LIVES OF MANY PEOPLE. NOT SURE EXACTLY WHY THIS EVENT OCCURRED. HOWEVER; AS STATED ABOVE THE B737 WAS RESPONDING TO HIS OWN RA AND THAT IS WHAT CAUSED OURS. THERE COULD BE NUMEROUS REASONS FOR HIS RA; SUCH AS ANOTHER ACFT ALTDEV OR A LACK OF SEPARATION ERROR BY ATC. UNSURE HOW TO AVOID THIS EVENT IN THE FUTURE. ONE THING I KNOW FOR SURE IS THAT TCAS SAVED US FROM DISASTER. I WOULD RECOMMEND A POLICY TO NEVER FLY WITHOUT TCAS BEING OPERATIVE. IT IS CURRENTLY ALLOWED TO BE MEL'ED.