PD vs Triforce on aatrox

To preface this I am not good at league math. That being said, given that the two builds for aatrox are the bork cleaver... and triforce steraks, in the triforce build why is triforce being taken over phantom dancer?

From what I can see, they are fairly comparable. After 2100 health, the 12% damage reduction is more effective health than the 250 from triforce. The movespeed passive makes up for the loss of phage ms stickiness, and given that aatrox has fairly long cds and uses them all at the beginning of an all in (sorta) the crit seems to be better with his inherent attack speed than the 2 sheen procs you get. I haven't done the math with calculations and rough dps, but just looking at it it would seem that PD is just better than triforce on him because of his lack of spellweaving, even when losing the 20% cdr.

Also bruiser items are the most expensive items in the game, and swapping triforce for PD would save 1100 gold.

Can somebody explain to me if I am correct, or if I am wrong what I am not taking into account? Or if somebody could do the math out that would also be great.