As per the above comments, the given solution is not correct.
From :
$∀x [(S(x) \land I(x)) \rightarrow \lnot H(x)]$
we have to start using the equivalence between $\forall$ and $\lnot \exists \lnot$ to get
$\lnot \exists x \lnot [(S(x) \land I(x)) \rightarrow \lnot H(x)]$.
Then, we have to apply the tautological equivalence between : $\lnot (p \rightarrow \lnot q)$ and $(p \land q)$ [you can check it with a truth-table] and convert the above formula into :
$\lnot \exists x [S(x) \land I(x) \land H(x)]$.
Example
Let $\Gamma$ the set of first-order Peano axioms: no variables free.
1) $\Gamma \vdash \exists x (x = 0)$ --- easily provable
2) $\Gamma, x=0 \vdash x=0$ --- obvious
3) $\Gamma \vdash x=0$ --- from 1) and 2) by $\exists$-elim : wrong !
4) $\Gamma \vdash \forall x (x=0)$ --- from 3) by $\forall$-intro,
1) $\Gamma, x=0 \vdash x = 0$
2) $\Gamma, x=0 \vdash \forall x (x=0)$ --- by $\forall$-intro: wrong !
3) $\Gamma \vdash x=0 \to \forall x (x=0)$ --- from 2) by $\to$-intro
4) $\Gamma \vdash \forall x [x=0 \to \forall x (x=0)]$ --- from 3) by $\forall$-intro
5) $\Gamma \vdash 0=0 \to \forall x (x=0)$ --- from 4) by $\forall$-elim.
The ground for the restriction on $\forall$-intro are linked to the "generalization principle":
what holds for any, holds for all.
Thus, in order to formalize this principle with a rule of inference, we read it as:
if something holds for an "arbitrary object", then it holds for all objects.
We have to capture the informal concept of “arbitrary object” by way of a syntactic criterion.
Consider now a variable $x$ in the context of a derivation: we shall call $x$ arbitrary if nothing has been assumed concerning $x$. In other words, $x$ is arbitrary at its particular occurrence in a derivation if the part of the derivation above it contains no hypotheses containing $x$ free.
Best Answer
Reading the sentences $\quad\lnot\forall v\:\,\lnot\phi\quad$ and $\quad\exists v\:\, \phi\quad$ literally:
It is false that every $v$ fails to satisfy $\phi.$
There is some $v$ that satisfies $\phi.$
Does this clarify their logical equivalence?