Theorem necon3ai

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon3ai.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	necon3ai	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		necon3ai.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
3	1 2	sylibr	⊢ ( 𝜑 → ¬ 𝐴 ≠ 𝐵 )
4	3	con2i	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 )