You are almost there. Instead of step 9, instantiate ∀yF(a(i)y) from step 3 with a(a(i)). The result would contradict step 8.
Im stuck and cant seem to find a way to continue
pr- ∀x(F(xa(x))→∼F(xx))
show ∴ ∼∃x∀yF(a(x)y
QN is disabled. Use UI, EI, and EG in the derivation
1. show ∼∃x∀yF(a(x)y
2. -------∃x∀yF(a(x)y) Ass id
3.--------∀yF(a(i)y) 2 ei
4.--------F(a(i)a(i)) 3 ui
5.--------∀x(F(xa(x))→∼F(xx)) pr
6.--------F(a(i)a(a(i)))→∼F(a(i)a(i)) 5 ui
7---------∼∼F(a(i)a(i)) 4 dn
8---------∼F(a(i)a(a(i))) 6 7 mt
9 --------∃x∼F(xa(x)) 8 eg
I dont know how to continue
Any advice will be appreciated, Thnx