Is the first occurrence of a variable bound?

Question

$ \forall \color{red}x (P(\color{blue}x) \rightarrow Q(\color{blue}x)) $

The occurrences of x in blue are bound by the quantifier $ \forall $, would the occurrence of x in red also be bound to $\forall$?

Is the x in red part of the formula which is 'universally quantified' or is it a part of the universal quantification itself?

Answer 1

The variable attached to the quantifier is considered part of the quantification and usually said to neither bound nor free; the concept of free vs. bound variables only applies to the formula that the quantifier has scope over. The variable that goes with the quantifier may be called a binding variable.

Answer 2

The red $x$ tells which symbol the quantifier is affecting.