complement of a property in the lambda calculus

Question

I'm trying to demonstrate that the complement of the complement of a property is equal to the property itself using $\lambda$ notation, i.e.

if $G=\lambda x(\neg Fx)$ and $R=\lambda x(\neg Gx)$, then R=F

I'm trying to derive this using $\beta$ and $\eta$ conversion, but I can't seem to find a proof.

Edit:

(1) $G=\lambda x(\neg Fx)$ (Assumption)

(2) $R=\lambda x(\neg Gx)$ (Assumption)

(3) $R=\lambda x(\neg (\lambda y(\neg Fy))x)$ (1, 2, relettering)

(4) $\lambda y(\neg Fy)y=\neg Fy$ ($\beta$ conversion)

(5) $\neg\lambda y(\neg Fy)=\neg\neg Fy$

(6) $Fy=\neg\neg Fy$

(7) $\neg\lambda y(\neg Fy)= Fy$

(8) $\neg\lambda y(\neg Fy)= \lambda y(Fy)$ ?

(9) $R=\lambda x(\lambda y(Fy))x$ (3, 8)

So if I can get to this point two steps of $\eta$ conversion gets me to the result. But I'm not sure whether this is correct...

Answer 1

The step with the $\beta$-conversion is not correct, or at least I do not understand what you mean by it. Maybe review the syntax of $\beta$-conversion? It applies only to terms of the form $$(\lambda v. B)\ T$$ but you seem to have applied it in some other way.

Also, where did your $R=\dots$ go?