We have Statements that are either true or false: eg. $4=0$
We have Statement variables that are either true or false, their value is assigned beforehand: e.g. : $A$
We have Proposition formulas that contain finitely many statement variables which are combined with Logical Operators, after assigning each Statement variable a truth value it becomes a statement: $A$ and $A\wedge B$ and are Proposition formulas for example.
We use this Lemma to prove the equivalence of a set of pairs of Proposition-formulas. One of those pairs is $(A\Rightarrow B,\neg B\Rightarrow \neg A)$. This is the proof as stated in the script 
Can somebody expalin me how the Lemma $1.9$ is used here?
A example for contraposition would be, alternative definitions of injectivity:
$f(x)=f(y)\Rightarrow x=y$ is equivalent to $x\neq y \Rightarrow f(x)\neq f(y)$
So $f(x)=f(y)$ would be $A$ in this case, i.e. it is not a statement but a statement variable. Can somebody explain me what again what are the conditions for an Expression t be a statement variable?