Need help for a proof in propositional logic about the relevant parts of a valuation. i'm just starting to write mathematical proofs so i wanted to asked you if my proof is correct. Do you see any flaws? Are there any comments i missed to state? Perhaps you have some suggestions how to write it down more elegant? The proof is no big deal, i just want to get down with the business of writing proofs and practice a bit by working on relatively simple claims, like the following:
Sentence about the relevant parts of an evaluation: Let $H$ be an expression in propositional logic and $v_1,v_2$ valuations with $v_1(\alpha)=v_2(\alpha)$ for all propositional variables in $H$. If $V$ is an evaluation funtion of $H$ then it follows that $V(H,v_1)=V(H,v_2)$. 
Proof 


*

*Let $H$ be an atomic formula $H \equiv \alpha$, then it holds that $v_1(\alpha)=V(\alpha, v_1)$ and $v_1(\alpha)=V(H,v_1)$. Because $v_1(\alpha)=v_2(\alpha)$ it also holds that $v_2(\alpha)=V(\alpha, v_2)$ and $v_2(\alpha)=V(H,v_2)$. Thus one can clearly see that $V(H,v_1)=V(H,v_2)$. So the sentence about the relevant parts of an evaluation holds for expressions of the form $\alpha$.

*Let $H \equiv \neg H$. So it holds that $V(H, v_1)=V(\neg H, v_1)=non(V(H,v_1))$. If it is true that $V(H,v_1)=V(H,v_2)$, then $non(V(H,v_2))=V(\neg H, v_2)=V(H, v_2)$ is also true. Thus $V(H,v_1)=V(H,v_2)$.


Out of Step 1 and Step 2 one can conclude that the sentence about the relevant parts of an evaluation also holds for complex statements of the form $\neg H$.
Similar reasoning would apply for complex statements of the form $H_1$o$H_2$ with o $\in \{\lor, \land, \rightarrow, \leftrightarrow\}$, which i will leave out here.
Looking forward to your comments and/or answers!
 A: Long comment
The proof is correct; maybe, we can streamline it a little bit.
1) Assume that $H$ is $α$ atomic and we have: $v_1(α)=v_2(α)$.
Being $H$ atomic, we have $v_1(α)=V(α,v_1)$ and $v_2(α)=V(α,v_2)$.
Thus, we can conclude that: $V(H,v_1)=V(α,v_1)=v_1(α)=v_2(α)=V(α,v_2)=V(H,v_2)$, and the property holds for $H$ atomic.
For 2), we cannot write $H \equiv\lnot H$, but we have to say that $H$ is $\lnot J$ and that, by induction, we assume that the property holds for $J$. Thus: $V(H,v_1)=V(¬J,v_1)$. 
Then we have to recall that valuations have only two possible values: $\{ 0,1 \}$ or $\{ \text T, \text F \}$ and thus, we may write: $V(¬J,v_1)= \text {not-}(V(J,v_1))$, where the function "$\text{not-}$" switches the value of the valuation.
Resuming, we have: $V(H,v_1)=V(¬J,v_1)= \text{not-}(V(J,v_1))$. 
By induction hypotheses, we have that $V(J,v_1)=V(J,v_2)$ and thus:

$V(H,v_1)=V(¬J,v_1)= \text{not-}(V(J,v_1))=\text{not-}(V(J,v_2))=V(¬J,v_2)=V(H,v_2)$,

and the property holds for a complex statements $H$ of the form $¬J$.
