The only way to convert the expression into a proper abstract is by somehow binding the variable by a quantifier or [lambda],-operator, or replacing it with a designating singular term, proper name, object constant, or definite descriptor. There are only a handful of possibilities for redefining the relational property of being necessarily identical to a bound variable x, and none that is sufficient for Kripke's purposes in trying to prove that all identity relations are necessary.
Why not, in lieu of a sufficient argument to the contrary, expect instead that Leibniz's principle applies to all identities, regardless of how they are formulated, including those expressed by means of rigidly designative proper names or object constants and those expressed by means of nonrigidly designative definite descriptors? Why not countenance and consider the modality of such identities as xFx = xGx as well as a = b?
Indeed, some logicians have proposed that we syntactically and semantically reduce proper names to definite descriptors as a way of unpacking their referential meaning.
The argument ignores the contrary implications of identity relations expressed by means of nonrigid designators such as definite descriptors.
Significantly, the part Kripke leaves out of consideration, nonrigidly designative identities expressed by means of definite descriptors rather than proper names or object constants or bound variables, supports a counterargument whose conclusion immediately contradicts Kripke's claim that identity relations generally are necessary.