Theorem necon1bi

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon1bi.1	$$⊢A\ne B\to \phi$$
Assertion	necon1bi	$$⊢\neg \phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		necon1bi.1	$$⊢A\ne B\to \phi$$
2		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
3	2 1	sylbir	$$⊢\neg A=B\to \phi$$
4	3	con1i	$$⊢\neg \phi \to A=B$$