Consider Basic Law $V$:

$\hat x$$F$($x$)=$\hat x$$G$($x$)$\equiv$($\forall$$x$)($F$$x$$\equiv$$G$$x$)

At first glance, it seems to have the same form as Leibniz's law

$x$=$y$$\equiv$($\forall$$F$)($F$$x$$\equiv$$F$$y$) (if one substitutes '$x$' for '$y$' and one assumes '$x$' satisfies both '$F$ ' and '$G$')

Since it is claimed that the logic of Frege's Begriffsschrift is second-order logic without comprehension principles, if, given second-order logic without comprehension principles with the extra axioms

$x$=$x$

$x$=$y$$\equiv$($\forall$$F$)($F$$x$$\equiv$$F$$y$)

and a 'course-of-values' operator '$\hat x$' that gives, when applied to $F$, $G$, the first-order objects that satisfy $F$, $G$; can one derive Basic Law $V$? If not, what further assumptions must one make in order to derive Basic Law $V$ and are these further assumptions valid?