Theorem necon1bbid

Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008)

	Ref	Expression
Hypothesis	necon1bbid.1	$$⊢\phi \to \left(A\ne B\leftrightarrow \psi \right)$$
Assertion	necon1bbid	$$⊢\phi \to \left(\neg \psi \leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon1bbid.1	$$⊢\phi \to \left(A\ne B\leftrightarrow \psi \right)$$
2		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
3	2 1	bitr3id	$$⊢\phi \to \left(\neg A=B\leftrightarrow \psi \right)$$
4	3	con1bid	$$⊢\phi \to \left(\neg \psi \leftrightarrow A=B\right)$$