Can logical equivalence be simplified in terms of simpler logical operators

Question

I have been researching the net for an answer, but sadly to no avail. I know implication can be simplified into simpler operators using de Morgan's and associative laws (such as this post here demonstrates), but I have found no similar solution for equivalence.

The logical compound expression that I am trying to simplify is the following using only conjunction, disjunction and negation:

$$\neg(((A\rightarrow \neg B) \,\&\, C) \vee (\neg A\leftrightarrow\neg C)$$

Thank you very much taken for the explanation!

Answer 1

I suspect you're thinking too hard about this. Look back at how bi-implication is defined in the first place: by definition, $A\leftrightarrow B$ is equivalent to

$(A\rightarrow B) \,\&\, (B\rightarrow A)$.

So if we know how to "unpack" $\rightarrow$, we can similarly "unpack" $\leftrightarrow$.

Of course this gets lengthy, but that's why we use abbreviations like $\leftrightarrow$ in the first place.