Theorem necon2bd

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

	Ref	Expression
Hypothesis	necon2bd.1	⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )
Assertion	necon2bd	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon2bd.1	⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	1 2	syl6ib	⊢ ( 𝜑 → ( 𝜓 → ¬ 𝐴 = 𝐵 ) )
4	3	con2d	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )