Theorem necon2abid

Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon2abid.1	$$⊢\phi \to \left(A=B\leftrightarrow \neg \psi \right)$$
Assertion	necon2abid	$$⊢\phi \to \left(\psi \leftrightarrow A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon2abid.1	$$⊢\phi \to \left(A=B\leftrightarrow \neg \psi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1	necon3abid	$$⊢\phi \to \left(A\ne B\leftrightarrow \neg \neg \psi \right)$$
4	2 3	bitr4id	$$⊢\phi \to \left(\psi \leftrightarrow A\ne B\right)$$