B737 FLT CREW HAS TO DIVERT TO THEIR ALTERNATE DUE TO WX AND LOW FUEL STATE. EMER DECLARED.

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WHILE ENRTE TO ZZZ; WE RECEIVED HOLDING INSTRUCTIONS TO HOLD OVER XXXXX INTXN AT 15000 FT. DISPATCH GAVE US A DIVERT FUEL OF 6200 LBS; BUT WE CALCULATED OUR DIVERT FUEL TO BE 7600 LBS. WE DISCUSSED IT AND WERE BOTH COMFORTABLE WITH IT. THERE WAS FOG AT ZZZ AND THE RVR KEPT FLUCTUATING UP AND DOWN TO ABOVE AND BELOW MINIMUMS. WE DEPARTED THE HOLD ENRTE TO ZZZ WITH THE MINIMUMS ABOVE 700 FT RVR. AS WE GOT CLOSER TO THE ARPT; THE RVR DROPPED TO BELOW MINIMUMS. WE THEN HELD ON THE RWY 10 LOC AT 2000 FT. WE TOLD ATC TO START WORKING ON A CLRNC FOR US TO GO TO ZZZ1 BECAUSE WE WERE GETTING CLOSE TO OUR DIVERT FUEL. WE WERE CLRED FOR THE APCH; AND STARTED INBOUND; BUT THE RVR DROPPED TO 600 FT. WE WERE AT OUR DIVERT FUEL OF 7600 LBS AND INFORMED ATC WE WANTED TO DIVERT TO ZZZ1. THEY STARTED VECTORING US TO THE WRONG ARPT. WE TOLD THEM WE WANTED TO GO TO ZZZ1 AND NEEDED A HIGHER ALT TO SAVE FUEL. ATC TOLD US TO CLB TO 3000 FT AND WERE VECTORING US TO A 090 DEG HDG. WE REPEATEDLY REQUESTED ON COURSE ZZZ1 AND A HIGHER ALT. WE TOLD THEM WE WERE GOING TO HAVE TO DECLARE MINIMUM FUEL; AND ATC SAID THAT THERE WAS TFC ABOVE US. ATC HAD US CLB TO 13000 FT AND KEPT US THERE; SO WE KEPT ASKING FOR HIGHER. AT THE ALT WE WERE AT; WE WOULD HAVE LANDED AT ZZZ1 WITH UNDER 3000 LBS; SO WE DECLARED EMER FUEL. THEN ATC HAD US CLB TO FL280. AT FL280 WE WOULD LAND WITH OVER 3000 LBS. WE TOLD ATC AND LANDED AT ZZZ1 WITH 3500 LBS. THE PROB WAS THAT ATC STARTED VECTORING US TO THE WRONG ARPT AND KEPT US AT A LOW ALT FOR A VERY LONG TIME. SUPPLEMENTAL INFO FROM ACN 685344: WHILE HOLDING AT THE INTXN AT 15000 FT; REVIEWED THE DIVERSION CHKLIST AND COORD WITH DISPATCH FOR A DIVERT TO ZZZ1 AT 7600 LBS OF FUEL (PERSONAL MINIMUM).