It appears philosophically unobjectionable to instantiate the universal reflexivity of identity in K(Q-W) assumption (2), [for all]xx = x, in the counterpart definite descriptor expression, xFx = xFx.
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.
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.