Theorem mtt

Description: Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001) (Proof shortened by Wolf Lammen, 12-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		biimt	$$⊢\neg \phi \to \left(\neg \psi \leftrightarrow \left(\neg \phi \to \neg \psi \right)\right)$$
2		con34b	$$⊢\left(\psi \to \phi \right)\leftrightarrow \left(\neg \phi \to \neg \psi \right)$$
3	1 2	bitr4di	$$⊢\neg \phi \to \left(\neg \psi \leftrightarrow \left(\psi \to \phi \right)\right)$$