It looks like you're trying to prove a substitution lemma, or something involving the substitution lemma (I'm assuming I understand the notation you're using). If I recall correctly, the proof proceeds by induction on the structure of the WFF. So, you will assume the truth of the antecedent, i.e. that for each p ∈ At(A), that v(p) = v'(p), then show the truth for each case of the WFF given the assumed truth of the antecedent and the rules for substitution....I think. Happy to be corrected if wrong.