FLC OF A B737 STOPPED DSCNT TO ASSIGNED ALT DURING ARR VECTORS DUE TO TCASII ALERT AND THE SIGHTING OF A B767 TURNING TOWARD THEIR ACFT 1400 FT BELOW. THE B767 CREW OBSERVED THE RPTR'S B737 AND ASKED APCH CTL IF THERE WAS A PROB. DURING TELECON AFTER LNDG; THE CTLR SUPVR ADVISED OF NO PROB SINCE EACH ACFT HAD THE OTHER IN SIGHT.

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FLT FROM SLC TO LAX IN CONTACT WITH SOCAL APCH FREQ 128.22 INBOUND CIVET 4 ARR PDZ VOR VICINITY. CLRNC TO DSND TO FL190. NEXT CLRNC; FIRST SLOW TO 250 KTS THEN DSND 16000 FT. NEXT CLRNC; TURN L HDG 180 DEGS DSND 14000 FT FOR 'SPACING.' AT THAT POINT WE WERE COMPLYING WITH PREVIOUS CLRNC. I READ BACK NEW CLRNC 'TURNING L 180 DEGS; DSND 14000 FT;' AND ADDED; 'WE'RE APPROX 235 KTS BECAUSE HE SAID 'FOR SPACING'.' AT THAT POINT IN THE TURN I IMMEDIATELY NOTICED AN ACFT OUT MY L SIDE WINDOW HDG NNW IN A L TURN TO A WESTERLY HDG BELOW AND ABOUT 5-10 MI FROM US. I CHKED TCASII AND IT SHOWED THE ACFT 1700 FT BELOW OUR ALT. WE WERE DSNDING THROUGH 15700 FT FOR 14000 FT AS INSTRUCTED. AT THAT POINT I INSTRUCTED MY FO FLYING TO LEVELOFF AND HE DID AT APPROX NO LOWER THAN 15400 FT. AT THAT POINT; WE HEARD A FLT COMMENT ON FREQ 'WHAT'S THIS DOING?' APCH RESPONDED 'MAINTAIN PRESENT ALT.' AFTER BEING TURNED BACK TOWARDS THE ARPT; I HAD TO ASK APCH IF THEY WANTED US AT THIS ALT (15400 FT) BECAUSE THEY NEVER ASSIGNED US A NEW ALT FROM 15400 FT. WHEN THE FLT RECEIVED FREQ CHANGE HE ASKED 'WAS THERE A PROB BACK THERE?' AND THE CTLR RESPONDED 'WE'RE LOOKING INTO IT AND LISTENING TO THE TAPES.' ON ARR LAX; I CALLED APCH ON THE PHONE AND TALKED TO SUPVR ON DUTY. HE INDICATED HE WAS STANDING NEXT TO THE CTLR THAT HANDLED OUR FLT AND TO STAND BY ONE. HE COVERED THE PHONE AS HE DISCUSSED WITH CTLR HIS VERSION. HE RETURNED BY SAYING 'OK; I'M IN BTWN HERE; LET'S HEAR YOUR SIDE.' I INDICATED WE WERE INITIALLY CLRED ON THE ARR TO FL190. THEN SLOW TO 250 KTS FIRST; THEN DSND TO 16000 FT. THEN TURN L TO A HDG OF 180 DEGS DSND TO 14000 FT. I TOLD HIM IN THE TURN I IMMEDIATELY SAW THE B767. AT THAT POINT HE SAID 'THAT'S WHAT I WANTED TO HEAR. THE FACT THAT BOTH OF YOU HAD VISUAL CONTACT ON EACH OTHER.' WE WERE NEVER CLOSER THAN 1400 FT VERTLY. AT THAT POINT OF THE CONVERSATION; THE SUPVR SEEMED SATISFIED AND WE HUNG UP.