Theorem necon3bd

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon3bd.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
Assertion	necon3bd	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon3bd.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
2		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
3	2 1	syl5bi	⊢ ( 𝜑 → ( ¬ 𝐴 ≠ 𝐵 → 𝜓 ) )
4	3	con1d	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )