ACR X TCASII RA DSND FROM ACFT VIS SEPARATION IN USE HAD NMAC WITH VFR ACFT BELOW. EVASIVE ACTION TAKEN. SEE AND AVOID CONCEPT.

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ACR X FROM SYR TO LGA WAS BEING RADAR VECTORED FOR AN LDA 'A' RWY 22 APCH INTO LGA ARPT. WE WERE LEVEL AT 4000 FT AND GIVEN A TA ABOUT A VFR ACFT AT 1 O'CLOCK AND I BELIEVE AT 3000 FT CLBING AND ALSO CONVERGING. THIS ACFT WAS NOT CURRENTLY TALKING TO THE CTLR WHO WAS WORKING OUR FLT. AS WE SEARCHED FOR THIS ACFT WE WERE THEN GIVEN A CLRNC TO DSCND TO 3000 FT AND SHORTLY AFTER THAT WE WERE GIVEN THE APCH CLRNC TO RWY 22. THE VFR ACFT THEN CHKED IN WITH OUR CTLR AND WAS ADVISED OF OUR POS. THE ACFT RESPONDED HE HAS US INSIGHT AND WAS INSTRUCTED TO MAINTAIN VISUAL SEPARATION FROM US. SECONDS LATER WE GOT A TA ON THE TCASII SYS WHILE WE WERE DSNDING AND HE WAS CLBING AND COMING AT US AT ABOUT OUR 2 O'CLOCK POS. THE TA SHOWED THIS ACFT TO BE ABOUT 1 1/2 TO 2 MI; 300 FT BELOW US AND CLBING IN OUR DIRECTION. A SPLIT SECOND LATER WE GOT A RA ON THIS ACFT. RA INSTRUCTED A DSCNT OF 1500 FPM. I STARTED DOWN AND THEN THE RA INSTRUCTED AN INCREASE IN RATE TO ABOUT 2000 FPM AS WE WERE PASSING ABOUT 3300 FT IN OUR DSCNT. WE HAD TO PASS THROUGH OUR ASSIGNED 3000 FT ALT BY ABOUT 300 FT BEFORE THE TCASII RESOLVED THE CONFLICT ON THIS ACFT. BY THIS TIME WE HAD DSNDED TO AN ALT OF ABOUT 2700 FT WHEN AT A SPLIT SECOND WE GOT AN RA ON TCASII TO CLB AT A RATE OF ABOUT 2000 FPM. AS I DID SO AND LOOKED OUT THE L SIDE I SAW AN ATR 42 ACFT FLY UNDER US L TO R AT ABOUT ONLY 200 FT TO 300 FT SEPARATION. I BELIEVE THE ATR 42 WAS VFR; TRANSITIONING UNDER THE LATERAL FLOOR OF THE CLASS B AIRSPACE AT 2500 FT; CUTTING ACROSS THE APCH CORRIDOR ENRTE TO EWR ARPT. WHEN WE GOT THE INSTRUCTION TO INCREASE DSCNT ON THE FIRST I WAS A LITTLE HESITANT TO DO SO. IF I HAD WENT TO THE FULL 2000 FPM INCREASE (I ONLY ATTAINED ABOUT 1800 FPM BY THE TIME FIRST CONFLICT WAS RESOLVED) I DO BELIEVE WE WOULD HAVE HIT THE ATR.