NMAC TURNED INTO A POTENTIAL CONFLICT AS TCASII ALERT STARTS PROVIDING SOME SEPARATION BTWN XING TFC; ABOUT SAME ALT. B747 VERSUS GA ACFT.

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ON DEP FROM SFO; UNDER CTL OF ZOA; HAD A NEAR MISS WITH A LIGHT ACFT. AFTER SEVERAL TFC CALLS FROM CTR TO BOTH ACFT; THE LIGHT ACFT FINALLY SAW US AND BROADCAST 'WE'RE ON YOUR NOSE.' WE NEVER SAW THE OTHER ACFT. THE TFC HAD PREVIOUSLY ASKED CTR 'WE'RE NOT ON A COLLISION COURSE; ARE WE?' THE TFC PASSED BY ESSENTIALLY CO-ALT AND LESS THAN 1/4 MI HORIZLY; ESTIMATED BY TCASII INDICATIONS. TCASII ANNOUNCEMENTS WERE 'TFC; TFC' FOLLOWED BY 'MONITOR VERT SPD;' ACCOMPANIED BY AN INDICATION TO CLB AT 1500- 2000 FPM (GREEN BAND ON VSI). CAPT (PF) HAD ALREADY STARTED A R TURN WHEN CTR FINALLY GAVE US A R TURN OF 20 DEGS. HAD WE WAITED TILL THEN; IT WOULD HAVE PROBABLY BEEN TOO LATE; SIT COULD PROBABLY HAVE BEEN AVOIDED BY AN EARLIER VECTOR AWAY FROM TFC. SUPPLEMENTAL INFO FROM ACN 344432: AFTER RECEIPT OF A CLB CLRNC FROM 5000-10000 FT FROM ZOA WE WERE GIVEN A TA OF A LIGHT ACFT AT 1 O'CLOCK 1 MI OPPOSITE DIRECTION OF FLT. WE ACQUIRED THE TFC ON TCASII AT ABOUT 7-8 MI MOVING FROM 1 O'CLOCK TO 11 O'CLOCK AS WE CLOSED. THE OTHER ACFT SAW US AND ASKED ZOA IF WE WERE ON A COLLISION COURSE? OAK RESPONDED 'NEGATIVELY.' ALL 3 OF US WERE NOT ABLE TO SEE THE TFC. AN ORDER TO CLB FROM THE TCASII. I BANKED SHARPLY R; WENT TO FULL PWR AND CLBED AT 2200 FPM TO AVOID THE TARGET. AFTER I STARTED THE MANEUVER I HEARD ZOA ISSUE A 20 DEG R TURN TO AVOID TFC; THIS ORDER WAS TOO LAKE; IN MY OPINION; BY A LONG WAY. THIS WHOLE SIT COULD HAVE EASILY BEEN AVOIDED BY PROMPT CTR ACTION AFTER WE REPEATEDLY ASKED FOR THE TFC'S POS. WE OBVIOUSLY DIDN'T SEE IT AND SHOULD HAVE BEEN VECTORED AROUND IT.