If I use an element chasing proof, can I prove the identity of two sets false by proving the negation of that identity?

For example, if A=(B∖C) is true, the following are true:

1. ∀x(x∈A→x∈B∧~x∈C)

2. ∀x(x∈B∧~x∈C→x∈A)

(A=(B∖C) is just an arbitrary example; I don’t really know its truth value)

But this sounds really just like a biconditional to me, if we take

∀x(x∈A) as P

∀x(x∈B∧~x∈C) as Q

then the truth of A=(B∖C) just means P↔Q. So surely if I want to prove A=(B∖C) is false, I can treat it as ~(P↔Q), which is the same as (P→~Q)∨(Q→~P).

Could anyone confirm if I am right please?