Given the following argument:
(i) $(\forall_x)(S(x)\rightarrow N(x))$,
(ii) $(\forall_x)(V(x)\rightarrow\lnot N(x))$,
(C) $(\forall_x)(V(x)\rightarrow\lnot S(x))$
The proof is given as Assume $V(x)$. By (ii), $\lnot N(x)$. By (i), $\lnot S(x)$. Thus, $\lnot S(x)$ follows from $V(x)$, and the conclusion holds.
However, given the following argument:
(i) $(\exists_x)(G(x)\rightarrow C(x))$,
(ii) $(\exists_x)(S(x)\rightarrow\lnot N(x))$,
(C) $(\exists_x)(S(x)\rightarrow\lnot G(x))$
The same proof would not work. From what I gather, it is the rule of the contrapositive in the first proof that allows it to work? If this is the case, is there a law somewhere that states the same cannot apply to wffs over an existential quantifier?