Theorem necon4ai

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon4ai.1	$$⊢A\ne B\to \neg \phi$$
Assertion	necon4ai	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		necon4ai.1	$$⊢A\ne B\to \neg \phi$$
2		notnot	$$⊢\phi \to \neg \neg \phi$$
3	1	necon1bi	$$⊢\neg \neg \phi \to A=B$$
4	2 3	syl	$$⊢\phi \to A=B$$