A free variable $x$ in $\exists x x<(y\cdot y) \wedge\forall y \neg x\doteq (y+y)$ Is $x$ a free variable in $\exists x x<(y\cdot y) \wedge\forall y \neg x\doteq (y+y)$?
I'm asking because you can read $\exists x x<(y\cdot y) \wedge\forall y \neg x\doteq (y+y)$ as:


*

*$(\exists x x<(y\cdot y) )\wedge (\forall y \neg x\doteq (y+y))$? Or

*$\exists x (x<(y\cdot y) \wedge\forall y \neg x\doteq (y+y))$?
In case 1 $x$ is free since $x$ is free in $\forall y \neg x\doteq (y+y)$ and in case 2 $x$ is not free.
How to read this? What is the rule?
 A: In logic, there is no generally accepted precedence rule for determining how missing parentheses should be inserted.   The "formula" that you started with is simply not a well-formed formula.   
A: There are "standard" conventions in math log regarding the omission of parentheses, based on "precedence" between connectives.
Regarding quantifiers, the omission of parentheses can cause big troubles.
In Herbert Enderton, A Mathematical Introduction to Logic (2nd ed Harcourt - 2001), page 78, we have :

$\lnot$, $\forall$, and $\exists$ apply to as little as possible. For example,

$\forall x \alpha \rightarrow \beta$ is $(\forall x \alpha \rightarrow \beta)$, and not $\forall x (\alpha \rightarrow \beta)$.


According to the above convention, the "correct" reading of your formula is 1).
The same convention is used in :
Stephen Cole Kleene, Mathematical Logic (1967), page 80 : "The quantifiers acts as unary operators in building formulas, and with our other unary operator $\lnot$ are ranked last under the convention for omitting parentheses."
in Dirk van Dalen, Logic and Structure (5th ed - 2013), page 58 : "We agree that quantifiers bind more strongly than binary connectives."
and in :
George Tourlakis, Lectures in Logic and Set Theory. Volume 1 : Mathematical Logic (2003), page 17 : "To minimize the use of brackets in the metanotation we adopt standard priorities of connectives: $\forall$, $\exists$, and $\lnot$ have the highest".
But a good advice is : "do not save ink !"
