Theorem necon2ad

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon2ad.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )
Assertion	necon2ad	⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon2ad.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )
2		notnot	⊢ ( 𝜓 → ¬ ¬ 𝜓 )
3	1	necon3bd	⊢ ( 𝜑 → ( ¬ ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )
4	2 3	syl5	⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )