Socrates' argumen,t "if the god-loved and the pious were the same, my dear Euthyphro, then if the pious was being loved because it was pious, the god-loved would also be being loved because it is god-loved" (Euthyphro, 11a), is in response to Euthyphro's second definition for piety that claims that the pious is the same as the god-loved (Euthyphro, 9e). Euthyphro doesn't realize it, but his definition for piety is an identity claim, or at least Socrates constructs it as one. Socrates is arguing that the pious is not the same as the god-loved. His argument says that if they were indeed the same, we would be able to substitute "pious" for "god-loved" in a statement without changing the truth-value of the statement in which the substitution is taking place. .
Here is a simple example of an identity claim: the author of The Apology is the author of The Republic. When I say that the 1) author of The Apology was a student of Socrates and a teacher of Aristotle, it is identical to saying that the 2) author of The Republic was a student of Socrates and a teacher of Aristotle. Notice how the truth-value remains the same. I should be able to do the same thing with the words "pious" and "god-loved" as I did with the expressions "author of The Apology" and "author of The Republic." But when the word "pious" is substituted for the word "god-loved," you end up changing the truth-value of the statement. For example, Euthyphro agrees that 1) the pious is being loved by the gods because it is pious is true (Euthyphro 10e), but disagrees that 2) the pious is being loved by the gods because it is god-loved is true (Euthyphro 10e). By substituting "pious' with "god-loved" we ended up with a false statement at 2, and all in accordance to beliefs Euthyphro holds.
Socrates' argument undermines Euthyphro's definition for piety by showing Euthyphro that if he holds his definition as true, then he is also saying that he believes that the pious is being loved by the gods because it is god-loved.