Theorem notbi

Description: Contraposition. Theorem *4.11 of WhiteheadRussell p. 117. (Contributed by NM, 21-May-1994) (Proof shortened by Wolf Lammen, 12-Jun-2013)

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

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3		id	$$⊢\left(\neg \phi \leftrightarrow \neg \psi \right)\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
4	3	con4bid	$$⊢\left(\neg \phi \leftrightarrow \neg \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
5	2 4	impbii	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \psi \right)$$