Theorem necon2bbid

Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon2bbid.1	$$⊢\phi \to \left(\psi \leftrightarrow A\ne B\right)$$
Assertion	necon2bbid	$$⊢\phi \to \left(A=B\leftrightarrow \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon2bbid.1	$$⊢\phi \to \left(\psi \leftrightarrow A\ne B\right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1 2	bitr3di	$$⊢\phi \to \left(A\ne B\leftrightarrow \neg \neg \psi \right)$$
4	3	necon4abid	$$⊢\phi \to \left(A=B\leftrightarrow \neg \psi \right)$$