Eequivalences rules:
This conveys a meaning that is actually much simpler so than you would think on first inspection.
Hence we can justify this, by using the following chain of rewrite steps based on the equivalences we've stated above that the:
1. By using the double negation rewrite: P => ¬¬P
(A ↔ B) ^ (¬A ?¬¬B)
2. By using De Morgan's Law: ¬P ^ ¬ Q => ¬(P? Q)
(A ↔ B) ^¬ (A? ¬B)
3. By using the commutativity of :? P? Q => Q? P
(A ↔ B) ^ ¬ (¬ B? A)
4. By using 'replace implication' from right to left: ¬ P? Q => P→ Q
(A↔ B) ¬ (B→ A)
5. By using 'replace equivalence' from left to right: P ↔ Q => (P→ Q)^ (Q → P)
((A → B)^ (B→ A)) ^ ¬ (B → A)
6. By using the associativity of : ^(P^ Q) R => P (Q^ R)
(A→ B) ^ ((B→ A) ^ ¬ (B → A))
7. By using the consistency equivalence above: P ^ ¬P => False
(A ↔ B) ^ False
8. By using the definition of ^ : False
There what does this mean? Like if it mean that our original sentence was always false: and there are no models that would make this sentence true. Such the another way to think about this is that the original sentence was too inconsistent with the rules of propositional logic there.