An equivalent definition for set equality using tautology

Question

We known from logic that, $$\left(p\Leftrightarrow q\right)\Leftrightarrow\left(\sim p \Leftrightarrow \sim q\right)$$ is a tautology, so when we see the definition of equality of sets, $$\left(A = B\right) \Leftrightarrow \forall x \left(x\in A \Leftrightarrow x\in B\right)$$ we have an iff statement, so using our tautology should be possible.

I'm trying to figure out if the next proposition is an equivalence definition for the equality of set.

$$\left(A = B\right) \Leftrightarrow \forall x(x\not\in A \Leftrightarrow x\not\in B)$$

We get that proposition negating both $\in$ from the original definition.

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so my questions are:

1. Is the last proposition true?
2. And if is not, why is it?

Answer 1

This is true, and could be a good exercise for you. Take my comment (below) as a sketch proof, and see if you can work through the details.

If $A=B$ then every element not in $A$ is not in $B$, and every element not in $B$ is not in $A$. For the converse, if every element not in $A$ is not in $B$ and every element not in $B$ is not in $A$, then it is true that $A=B$.

Note that $A = \{1, 2\}$ and $B = \{1\}$ is not a valid counterexample. Try to understand why --- this will help with your own proof.