GA SMA ENCOUNTERED ICING WHILE FLYING AN IFR TRAINING FLT AND HAD TO REQUEST RADAR VECTORS TO VFR CONDITIONS.

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NBOUND ON V21 IFR TRNING FLT IN IFR CONDITIONS CLRED FOR POCATELLO ILS RWY 21 APCH. AFTER XING POCATELLO VOR TURNED RIGHT TO TRACK APR TO IAF TYHEE. INCORRECT READING ON DG; WE WERE NOT ABLE TO IDENT LOM. LEFT TURN BACK TO POCATELLO VOR AND ADVISED SLC CENTER. CENTER CLRED US TO HOLD. SW POCATELLO VOR R-235 LEFT TURNS. 8000' ASKED SLC CENTER FOR CLRNC TO BURLEY VIA V-269 CLB TO VFR AND WE WOULD CANCEL IFR UPON REACHING VFR CONDITIONS. SLC CENTER CLRED US AS WE ASKED; CLB 10000'. AS WE STARTED TO CLB; FLEW INTO HEAVY FREEZING RAIN. LOST AIRSPD AND ALSO COULD NOT CLB. ADVISED CENTER WE HAD ICE AND COULD NOT CLB. TURNED ON PITOT HEAT; REGAINED AIRSPD. I BELIEVED AT THAT TIME CENTER ASKED US IF WE WOULD LIKE TO SHOOT THE APCH AT POCATELLO ILS RWY 21 AGAIN. AT THAT TIME AN AIRLINE CAPT ADVISED HE HAD VFR CONDITIONS BELOW HIM. CENTER ASKED US IF WE WANTED TO SHOOT POCATELLO ILS 21 APCH OR RADAR VECTORS TO VFR CONDITIONS AS ADVISED BY THE AIRLINE CAPT. WE ELECTED TO TAKE RADAR VECTORS TO VFR CONDITIONS BECAUSE OF THE ICE. NOT BEING ABLE TO MAINTAIN ALT WE DECLARED AN EMER. NOT FINDING VFR CONDITIONS WE HAD SEVERAL DIFFERENT READINGS ASSIGNED. WE ASKED FOR RADAR VECTOR TO THE LOWEST TERRAIN. ALT DROPPED AS LOW AS 5700' MSL. AT THAT POINT WE HAD SOME ICE MELT AND WE WERE ABLE TO CLB TO 6100'. SLC CENTER SAID HE HAD CALLED BLACKFOOT; A VFR ARPT; SAID GND PERSONNEL RPTED 500' CEILING 1 1/2 MI VIS. COULD GIVE US RADAR VECTOR OVER THE ARPT OR WE COULD HAVE RADAR VECTOR TO IDAHO FALLS LOC BC APCH RWY 2. WE TOOK LOC BC RWY 2 IDAHO FALLS; LANDED IDAHO FALLS RWY 2. THE PROBS THAT CAUSED THE SITUATION: INADEQUATE PREFLT IN REGARDS TO WX; ICING; BRIEFING AND PLANNING. SHOULD HAVE FLOWN THE POCATELLO APCH AGAIN; SHOULD HAVE FILED AN ALTERNATE ARPT ON FILE PLAN; SHOULD HAVE EXECUTED A 180 DEG TURN ONCE ICE STARTED TO DEVELOP. SHOULD HAVE FOLLOWED THE WX SITUATION BETTER ENRTE AND AT THE DEST ARPT.