P50 Controller described conflict event; alleging pilot was most likely fixated on TCAS equipment in lieu of listing to ATC instructions and visually observing potential traffic.

Narrative

Aircraft X was on a vector northbound at 4;000. Aircraft Y called me off SDL for a transition over PHX to CGZ. Initially I climbed Aircraft Y to 5;000 assuming he would want 5;500 (as most do). When I asked his requested altitude; he said 3;500. Since he was just leaving 4;000; I said descend and maintain 3;500. I called traffic over PHX northbound at 4;000; six miles away. I exchanged the traffic with Aircraft X as well. Aircraft Y did not descend. I had to take action to separate these two aircraft that had 6 miles to be separated; but because Aircraft Y did not listen to my instruction and do what I said; I had to turn Aircraft X to a 270 heading. It was an unusual heading and the instructor in the aircraft questioned me which was making the situation worse. I said in plain 'easy to understand' language to turn westbound NOW! Once I regained separation; I questioned Aircraft Y as to why he did not descend and was indicating 4;100. His answer was that he saw traffic 100 FT below him. Based on that response; I believe the pilot was fixated on his TCAS or some display in the cockpit and not trusting and/or following instruction. When I initially called the traffic; he should have heard the traffic was 6 miles away at 4;000; northbound. Pilots need to trust ATC. This lack of trust;lack of following instruction almost caused a mid-air. Now when I turned Aircraft X out of the way; I came within 1.5 miles of freeways boundary. I pointed Aircraft X out to freeway late. It is virtually impossible to point out in time; since the corridor we have to vector aircraft over PHX is only about 3.25 miles wide. That does not give us much 'wiggle-room'. On one side is another sector. On the other side is 'sterilized departure airspace' that we have to stay out of. Recommendation; get pilots to look out the window and not at their displays; especially if they are VFR! Also; get pilots to listen to controllers. If the pilot is unsure; ASK THE CONTROLLER.