This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ \neg φ \to (\neg ψ \leftrightarrow (ψ \to φ))$

Step	Hyp	Ref	Expression
1		biimt	$⊢ \neg φ \to (\neg ψ \leftrightarrow (\neg φ \to \neg ψ))$
2		con34b	$⊢ (ψ \to φ) \leftrightarrow (\neg φ \to \neg ψ)$
3	1 2	bitr4di	$⊢ \neg φ \to (\neg ψ \leftrightarrow (ψ \to φ))$