Theorem necon3bi

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon3bi.1	⊢ ( 𝐴 = 𝐵 → 𝜑 )
Assertion	necon3bi	⊢ ( ¬ 𝜑 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon3bi.1	⊢ ( 𝐴 = 𝐵 → 𝜑 )
2	1	con3i	⊢ ( ¬ 𝜑 → ¬ 𝐴 = 𝐵 )
3	2	neqned	⊢ ( ¬ 𝜑 → 𝐴 ≠ 𝐵 )