Theorem necon4bbid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012)

	Ref	Expression
Hypothesis	necon4bbid.1	$$⊢\phi \to \left(\neg \psi \leftrightarrow A\ne B\right)$$
Assertion	necon4bbid	$$⊢\phi \to \left(\psi \leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon4bbid.1	$$⊢\phi \to \left(\neg \psi \leftrightarrow A\ne B\right)$$
2	1	bicomd	$$⊢\phi \to \left(A\ne B\leftrightarrow \neg \psi \right)$$
3	2	necon4abid	$$⊢\phi \to \left(A=B\leftrightarrow \psi \right)$$
4	3	bicomd	$$⊢\phi \to \left(\psi \leftrightarrow A=B\right)$$