I was following the solution of one particular propositional logic puzzle where I encountered that the initial well-formed formula was reduced to:
((Q → ((A → B) ∧ (¬A → ¬B))) ∧ (¬Q → ((¬A → B) ∧ (A → ¬B))))
The example then used the definition of Logical Equivalence(<=>) and the below identity:
((¬p ↔ q) ↔ ¬(p ↔ q))
to reduce the formula to
Q ↔ (A ↔ B)
Can anyone please help me understand how the final concise form was derived using the identity?
Another equivalence you want to use here is:
$P \leftrightarrow Q \Leftrightarrow (P \to Q) \land (\neg P \to \neg Q)$
Thus, the $(A \to B) \land (\neg A \to \neg B)$ part in your original expression is equivalent to $A \leftrightarrow B$.
Likewise, the $(\neg A \to B) \land (A \to \neg B)$ part is equivalent to $A \leftrightarrow \neg B$
Hence, you get:
$(Q \to (A \leftrightarrow B)) \land (\neg Q \to (A \leftrightarrow \neg B))$
So by the provided equivalence, that is the same as:
$(Q \to (A \leftrightarrow B)) \land (\neg Q \to \neg (A \leftrightarrow B))$
And using the earlier mentioned equivalence yet again, that is equivalent to:
$Q \leftrightarrow (A \leftrightarrow B)$