Are these arguments (rules of inference) the same? The full question is:
If $m$, then $h$.
We have either $h$ or $w$.
Therefore, if not $h$, then I will not have $m$ and I have $w$.
Show that this above argument is logically valid.
Let $h,w,m$ be propositions. Argument 1 is
$$\begin{align}&m\rightarrow h \\ &h\vee w \end{align}$$ $$ \rule{2in}{.5pt}$$ $$\therefore\neg h \rightarrow (\neg m)\wedge w$$  
Argument 2 is
$$\begin{align}&m\rightarrow h\\ &h\vee w \\ &\neg h \end{align}$$ $$\rule{2in}{.5pt}$$$$\therefore (\neg m)\wedge w$$  
Are these saying the same thing? I'm trying to be 100% correct about this. The first two compound propositions were premises, and the question said
"If not $h$, then I will not have $m$ and I have $w$".  
So I'm not sure if I can say $\neg h$ is a premise (or assumption) from question, and this leads to $(\neg m) \wedge w$, meaning $\neg h \rightarrow (\neg m)\wedge w$ (this is for argument 2)  
The solution says argument 1 is the solution.  
 A: Rather, I'd say your "argument 2" is the step before proving "argument 1".
You take the premises $\bbox[lemonchiffon]{m\to h}$ and $\bbox[lemonchiffon]{h\vee w}$, and make assumption $\bbox[lemonchiffon]{\neg h}$ to conclude $\bbox[lemonchiffon]{\neg m\wedge w}$, then you discharge the assumption to demonstrate what was to be proven: that if the premises are justified then, $\bbox[lemonchiffon]{\neg h\to (\neg m\vee w)}$.

Since $\bbox[lemonchiffon]{{m\to h, \neg h\vdash \neg m}}$ (by modus tollens) and $\bbox[lemonchiffon]{{h\vee w, \neg h\vdash w}}$ (by disjunctive syllogism), therefore $ \bbox[lemonchiffon]{m\to h, h\vee w, \neg h~\vdash ~\neg m\wedge w}$ (by conjunctive introduction).   Since that, therefore $\bbox[lemonchiffon]{ m\to h, h\vee w~\vdash ~\neg h\to (\neg m\wedge w)}$ (by conditional introduction).   Thus demonstrating that the argument is valid.
$$\dfrac{m\to h\\ h\vee w}{\neg h\to (\neg m\wedge w)}$$
A: \begin{align}
\text{Given: }\qquad m &\rightarrow h \\ 
\therefore\quad \lnot\, h &\rightarrow \lnot\, m \\ 
\\
\text{Given: }\qquad h &\vee w \\
\therefore\quad \lnot\, h &\rightarrow w \\ 
\\
\text{taken together: }\quad\lnot\, h &\rightarrow \lnot\, m \land w \\
\end{align}
$\lnot\, h$ is not an outcome; the implications of $\lnot\, h$ are the outcome. However if you are given $\lnot\, h$ as a premise, you can then detach the implication from the statement and give those as an outcome. so:
\begin{align}
\text{Given: }\quad\lnot\, h &\rightarrow \lnot\, m \land w \\ 
\text{and: }\qquad\lnot\, h \\ 
\therefore\quad \lnot\, m &\land w \\ 
\end{align}
