If there is an interpretation $I$ in wich $A=\forall x (P(x)\rightarrow Q(x))$ is true, then $(P(x)\rightarrow Q(x))$ is not true ands is not false in $I$.
I need to justifiy this is false. So, if A is true for $I$ this means that every valuation on $I$ satisfies A. A little more formally, valuation $v$ satisfies $\forall x (P(x)\rightarrow Q(x))$ iff $v$ satisfies $P(x)\rightarrow Q(x)$ and every x-equivalent valuation $v'$ satisfies $P(x)\rightarrow Q(x)$. Therefore, it doesnt exists a valuation on $I$ in wich $P(x)\rightarrow Q(x)$ is false, then $P(x)\rightarrow Q(x)$ is true for $I$.
Is this ok?
I think you did a fine job with the proof.
So the only suggestion I'll make is simply a matter of "parsing"/choice of words, not directly related to the "content" of the proof. I'll italicize the word choices, then follow each italicization with a suggested replacement, which you can "take or leave."
"... [M]ore formally, valuation $v$ satisfies $\forall x (P(x)\rightarrow Q(x))$ iff [if and only if] $v$ satisfies $P(x)\rightarrow Q(x)$, and every x-equivalent valuation $v'$ satisfies $P(x)\rightarrow Q(x)$. Therefore, it doesn't exists [there does not exist] a valuation on $I$ in [for?] which $P(x)\rightarrow Q(x)$ is false, then [and hence it follows that] $P(x)\rightarrow Q(x)$ is true for $I$."