Quantifier explination: $\forall x(R\to Q(x))\equiv R \to \forall x (Q(x))$ I don't understand the proof for this problem and couldn't find the answer anywhere.
$$\forall x(R\to Q(x))\equiv R \to \forall x (Q(x))$$

Assume $\forall x(R\to Q(x))$. We want to prove $R\to \forall x Q(x)$.
So we assume $R$ and attempt to prove $\forall x Q(x)$.
So we let $x$ be arbitrary. For this $x$ we have, by specialization, that $R\to Q(x)$. As $R$, modus ponens gives us $Q(x)$. As $x$ was arbitrary, $\forall x Q(x)$. So we derived $\forall x Q(x)$ based on the hypthesis that $R$ holds. In other words, we have shown $R\to \forall x Q(x)$. This shows the $\implies$ direction of the desired equivalence.
Now assume instead that $R\to \forall x Q(x)$.
We want to prove $\forall x(R\to Q(x))$. 
So let $x$ be arbitrary. 
We want to prove $R\to Q(x)$.
So assume $R$. Then by modus ponens $\forall x Q(x)$.
In particular, $Q(x)$.
Thus we showed $Q(x)$ from $R$, i.e., we showed $R\to Q(x)$.
As $x$ was aribitrary, we showed $\forall x(R\to Q(x))$. This shows the other direction.
