Here's my attempt so far in proving the equivalence:
The question says "x does not occur freely in A" (which means it is either the variable immediately following a quantifier or it is in the scope of some quantifier involving x). Let I be an interpetation with domain on D, and v be an I-assignment. Suppose . We wish to show that . I don't know if this is correct but I'm considering two cases depending on wether holds for all :
Case 1: holds for all . So so . Is this correct?
Case 2: There is some such that does not hold... And we assumed that there is a d such that . But how can I complete this part?
I greatly appreciate any help on how to prove this question...