Theorem con2bi

Description: Contraposition. Theorem *4.12 of WhiteheadRussell p. 117. (Contributed by NM, 15-Apr-1995) (Proof shortened by Wolf Lammen, 3-Jan-2013)

	Ref	Expression
Assertion	con2bi	$$⊢\left(\phi \leftrightarrow \neg \psi \right)\leftrightarrow \left(\psi \leftrightarrow \neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		notbi	$$⊢\left(\phi \leftrightarrow \neg \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \neg \psi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	2	bibi2i	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \neg \psi \right)$$
4		bicom	$$⊢\left(\neg \phi \leftrightarrow \psi \right)\leftrightarrow \left(\psi \leftrightarrow \neg \phi \right)$$
5	1 3 4	3bitr2i	$$⊢\left(\phi \leftrightarrow \neg \psi \right)\leftrightarrow \left(\psi \leftrightarrow \neg \phi \right)$$