LEAR35 FLT CREW REPORTS TCAS TA AT FL390 WITH B737. FLT CREW ELECTS TO CLIMB BACK TO FL400 AND NO RA IS GENERATED.

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I WAS CHKING ON THE PAX'S STATUS IN THE CABIN AND WAS SEATED JUST BEHIND THE MAIN CABIN DOOR. I FELT THE PLANE MAKE A NOSE DOWN ADJUSTMENT AND OBSERVED THE FO START A NORMAL DSCNT. SHORTLY AFTER THE DSCNT WAS INITIATED; I FELT ANOTHER PITCH CHANGE; THIS TIME IN THE UP DIRECTION; AND GLANCED FORWARD TO THE COCKPIT. I SAW THE FO START TO INITIATE A CLB AND REFLECT THE DEDICATED 'TFC' SCREEN ON THE MFD. AT THE SAME TIME I HEARD A 'TFC' CALLOUT FROM THE TCAS AND OBSERVED A YELLOW TARGET ON THE MFD. I QUICKLY TOOK MY SEAT IN THE COCKPIT ASSESSING THE SITUATION. THE ALT ALERTER WAS SET TO FL380; THE FO WAS CLBING THROUGH FL393; AND I HEARD ATC ISSUE AN ACR JET AN IMMEDIATE DSCNT CLRNC. I ASSUMED CTL AUTH 'I GOT IT;' CLICKED OFF THE AUTOPLT; AND INITIATED A 'HARD-UP' CLB WITH AN AGGRESSIVE PUSH AT THE END TO LEVEL AT FL400. I THEN IMMEDIATELY ANNOUNCED TO ATC THAT 'ACFT X IS LEVEL AT 400.' ATC CONFIRMED THIS AND ISSUED A 'MAINTAIN 400' CLRNC. UPON REVIEW WITH THE FO; HE WAS ISSUED A DSCNT CLRNC TO FL380 FROM FL400 AND ACKNOWLEDGED. HE COMMENCED A DSCNT AND WAS MIDWAY THROUGH WHEN HE GOT THE ALERT. A REVIEW OF EVENTS AND A CONVERSATION WITH ATC YIELDED THE FOLLOWING: ATC DID ISSUE THE DSCNT CLRNC IN ERROR; THEY DID APOLOGIZE AND MADE THE STATEMENT WE (THE CREW) DID NOTHING WRONG. THE CREW RESPONDED CORRECTLY TO A TCAS CALLOUT BY ACQUIRING THE TARGET VISUALLY; INITIATING AN ESCAPE MANEUVER; AND THEN INFORMING ATC OF THE DEV. I FEEL THE FLT CREW DID EVERYTHING RIGHT AND CAN'T THINK OF ANYTHING ON OUR END TO ADD TO AVOID THE EVENT. ATC MADE A MISTAKE FOR REASONS UNKNOWN AND THE ONBOARD EQUIP AND PROCS WORKED CORRECTLY TO ALLOW A SUCCESSFUL OUTCOME.