LGT HAS NMAC JUST OUTSIDE DTW TCA. ATC BLAMES 'STUCK' MIKE.

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THE FO WAS FLYING THE ACFT. THE CAPT WAS HANDLING ROUTINE COM. WHILE DSNDING TO 6000 FT ON THE APCH TO ILS 21R AT DTW; THE FO SAW AND ANNOUNCED AN INTRUDER ACFT AT ABOUT 1 O'CLOCK; XING R TO L IN FRONT OF OUR ACFT. THE INTRUDER ACFT WAS AT OUR ALT; OR SLIGHTLY BELOW US; WITH AN ESTIMATED 300 FT HORIZ SEPARATION. I TOOK COMMAND OF OUR ACFT AND INITIATED A L ROLL; TOGA CLB. THE EVASIVE MANEUVER RESULTED IN A 30-40 DEG HDG CHANGE AND AN APPROX 2000 FT CLB. WE WERE THEN GIVEN A NEW CLRNC AND CONTINUED THE APCH IN A NORMAL FASHION. THE INTRUDER ACFT WAS FLYING IN AND ABOUT THE SCATTERED CLOUD LAYER WITH BASES AT 6000 FT. WHEN WE QUESTIONED ATC AS TO WHY THE TFC WAS NOT CALLED; HE STATED THAT WE HAD JUST BEEN HANDED OFF TO HIM AND HE DIDN'T HAVE TIME TO TELL US. WHEN HE DID TRY TO TELL US; HE COULDN'T BECAUSE WE HAD A 'STUCK MIKE.' THE STUCK MIKE RESULTED WHEN I GRABBED THE SIDE-STICK CTLR TO INITIATE THE REQUIRED EVASIVE MANEUVER. THE MIKE TRIGGER IS LOCATED ON FRONT OF THE SIDE-STICK. THE FREQ HAD BEEN CLR UNTIL EVASIVE ACTION WAS INITIATED. THE FOLLOWING FACTS ARE EVIDENT: HAD WE BEEN IN IMC; THERE WOULD HAVE BEEN A COLLISION. HAD THE FO NOT SCANNED WHEN HE DID; THERE WOULD HAVE BEEN A COLLISION. THERE WERE NO INJURIES OR DAMAGE TO THE ACFT. I MADE 2 ANNOUNCEMENTS TO THE PAX EXPLAINING THE INCIDENT. THE FO WAS AT THE COCKPIT DOOR DURING PAX DEPLANING ANSWERING ANY QUESTIONS ASKED. THE PAX WERE CONCERNED; BUT CALM. WITH FEW EXCEPTIONS; EVERY ONE THANKED HIM FOR OUR ACTIONS. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR DESCRIBED THE SIDE STICK AND THE MIKE 'TRIGGER' USED TO KEY THE MIKE. DURING THE EVASIVE ACTION; THEY WENT NORDO BECAUSE HE HAD GRIPPED THE MIKE KEY DURING THE MANEUVER. AFTER CLR OF THE TFC; THE FO HAD STRONG WORDS FOR ATC. ATC SAID; 'WHY ARE YOU MAD AT ME?' CLAIMING THEIR MIKE WAS STUCK. CREW WONDERS WHAT GOOD A TA IS IF IT IS GIVEN WHILE THEY ARE IN THE PROCESS OF EVADING THE TFC.