A logical implication in which both sides are unrelated Is this proposition true or not:
$$ \forall x(x=x) \rightarrow \forall x( x^2 -a^2 = (x+a)(x-a))$$
The left side $x=x$ does not have to do anything with the right side $x^2 - a^2 = (x+a)(x-a),$ but they are both true in any case; does this mean that a tautology is implying a tautology, and thus this statement is true?
 A: That the left-hand side has "nothing to do" with the right-hand side does not matter. Classical logic only cares about truth values, not contextual relevance or causal relation or anything.
The left-hand side is, as you suspect, tautological: There is no way to make it false, no matter which domain of discourse the quantifier ranges over.
The right-hand side is not always true. It is true with the standard interpretation of the symbols $+, -, \cdot, ²$, but false if we e.g. were to assign the function symbol $²$ the meaning "square root". When checking for tautologicity, such non-standard interpretations have to be considered as well!
An implication with a tautological antecedent is logically equivalent to the succedent, that is, if the left-hand side is a tautology, the implication as a whole will take the truth value of the right-hand side. Since $\forall x(x=x)$ is tautological, and $\forall x( x^2 -a^2 = (x+a)(x-a))$ is true in the "real world" but false for different interpretations of the symbols, the implication comes out as true, but not tautological.
A: Some definitions $\large($tautology $\subseteq$ validity $\subseteq$ truth$\large)$:

*

*a tautology is a formula that is true in its
truth-functional form regardless of interpretation;

*a validity is a formula that is true
regardless of interpretation;

*a (synthetic or analytic) truth is a formula that is true in a particular
interpretation.

Thus in the given statement $$\forall x \left(x=x\right)\rightarrow \forall x\forall y \left(x^2 -y^2 = (x+y)(x-y)\right),$$

*

*the antecedent (call it $L$) is valid (and therefore
true in the standard interpretation), but not tautological;

*the consequent (call it $R$) is true in the standard interpretation, but neither valid nor tautological;

*the statement as a whole is true in the standard interpretation
$\left(\text{so we can write }\,L\Rightarrow R\,\right),$ but neither
valid nor tautological.

P.S. I prefer to call $\,\large\rightarrow\,$ the material conditional and $\,\large\implies\,$ the implication symbol. (Why I don't call $\,\rightarrow\,$ “implication”.)
