Theorem notbinot1

Description: Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017)

	Ref	Expression
Assertion	notbinot1	$$⊢\neg \left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nbbn	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$
2	1	bicomi	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \psi \right)$$
3	2	con1bii	$$⊢\neg \left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$