Theorem imnot

Description: If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication ( ph -> ps ) , the other ones being ax-1 (true consequent), pm2.21 (false antecedent), pm5.5 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019) (Proof shortened by Wolf Lammen, 26-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		mtt	$$⊢\neg \psi \to \left(\neg \phi \leftrightarrow \left(\phi \to \psi \right)\right)$$
2	1	bicomd	$$⊢\neg \psi \to \left(\left(\phi \to \psi \right)\leftrightarrow \neg \phi \right)$$