Theorem necon1abid

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon1abid.1	$$⊢\phi \to \left(\neg \psi \leftrightarrow A=B\right)$$
Assertion	necon1abid	$$⊢\phi \to \left(A\ne B\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon1abid.1	$$⊢\phi \to \left(\neg \psi \leftrightarrow A=B\right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1	necon3bbid	$$⊢\phi \to \left(\neg \neg \psi \leftrightarrow A\ne B\right)$$
4	2 3	bitr2id	$$⊢\phi \to \left(A\ne B\leftrightarrow \psi \right)$$