Theorem necon1bd

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon1bd.1	$$⊢\phi \to \left(A\ne B\to \psi \right)$$
Assertion	necon1bd	$$⊢\phi \to \left(\neg \psi \to A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon1bd.1	$$⊢\phi \to \left(A\ne B\to \psi \right)$$
2		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
3	2 1	syl5bir	$$⊢\phi \to \left(\neg A=B\to \psi \right)$$
4	3	con1d	$$⊢\phi \to \left(\neg \psi \to A=B\right)$$