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 ≠ 𝐵 → 𝜑 )
Assertion	necon1bi	⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon1bi.1	⊢ ( 𝐴 ≠ 𝐵 → 𝜑 )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2 1	sylbir	⊢ ( ¬ 𝐴 = 𝐵 → 𝜑 )
4	3	con1i	⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 )