Theorem necon3bd

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon3bd.1	$$⊢\phi \to \left(A=B\to \psi \right)$$
Assertion	necon3bd	$$⊢\phi \to \left(\neg \psi \to A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon3bd.1	$$⊢\phi \to \left(A=B\to \psi \right)$$
2		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
3	2 1	syl5bi	$$⊢\phi \to \left(\neg A\ne B\to \psi \right)$$
4	3	con1d	$$⊢\phi \to \left(\neg \psi \to A\ne B\right)$$