NMAC AND ALT XING RESTRICTION NOT MET.

Narrative

I WAS IN COMMAND OF FLT; AN MLG; ENRTE FROM CVG TO LGA. MY FO WAS ACTUALLY MANIPULATING THE CTLS OF THE ACFT. WE WERE INBOUND TO LGA AT FL180 JUST E OF THE MILTON VOR. ZNY ISSUED US A CLRNC TO DSND NOW TO 15000 FT BUT CROSS VIBES AS PER THE MILTON 7 ARR AT 13000 FT. MY FO; USING THE AUTOPLT; BEGAN OUR DSCNT. I CALLED LEAVING FL180. AS WE DEPARTED APPROX 17500 FT; THE FO NOTED TFC ON OUR TCASII AT OUR 12 O'CLOCK POS; JUST INSIDE THE 10 MI TCASII RANGE. WE DO NOT RECALL A VERBAL WARNING OF 'TFC' FROM TCASII. I DO RECALL THAT THE TFC WAS BELOW US BUT CLBING AND CLOSING HEAD TO HEAD. I ASKED ATC IF THEY HAD TFC FOR US AT OUR 12 O'CLOCK POS. AT THIS POINT SEVERAL THINGS OCCURRED VERY FAST. AS I SAID; WE DO NOT REMEMBER A TA BEING GIVEN BY TCASII. THE FIRST TCASII WARNING WAS 'CLB; CLB.' ATC RESPONDED TO THE AFFIRMATIVE ABOUT HAVING TFC FOR US. AT ABOUT 16500 FT; IN RESPONSE TO TCASII; THE FO AND MYSELF BEGAN A SIMULTANEOUS PULL-UP TO REVERSE OUR DSCNT. I IMMEDIATELY ADVISED ATC THAT WE WERE RESPONDING TO A TCASII ALERT AND CLBING TO 17000 FT. ATC CONFIRMED THIS AND RECLRED US BACK TO 17000 FT. THE FO CONTINUED THE TCASII MANEUVER WHILE I STATED THAT I WAS 'OUTSIDE FOR THE INTRUDER.' AS WE CLBED THROUGH ABOUT 16800 FT; I SAW A LIGHT TWIN PROP ACFT FLASH UNDER OUR R WING WITH ZERO HORIZ CLRNC AND ABOUT 200 TO 300 FT OF VERT SEPARATION. I THEN RELATED TO ATC THAT WE HAD JUST AVOIDED ANOTHER ACFT BY ABOUT 300 FT. ALL OF THIS TOOK PLACE IN THE SPAN OF ABOUT 30 SECONDS. WITHOUT THE TCASII ADVISORY TO CLB; I FEEL I WOULD NOT BE WRITING THIS TODAY. ATC THEN SUBSEQUENTLY RECLRED US TO CROSS VIBES AT 13000 FT AS PER THE ARR. UNFORTUNATELY; DUE TO THE PREVIOUS EVENT; WE WERE UNABLE TO COMPLY AND WERE ABOUT 1000 FT HIGH WHEN XING THE FIX. I FEEL STRONGLY THAT THE UNSETTLING NATURE OF THE NEAR MISS CONTRIBUTED TO OUR NON COMPLIANCE. SUPPLEMENTAL INFO FROM ACN 222684: I ASKED THE CAPT TO GET INFO ON INTRUDER. CTR REPLIED THERE WAS A VFR TFC AT OUR 12 O'CLOCK AT 16500.