Theorem nebi

Description: Contraposition law for inequality. (Contributed by NM, 28-Dec-2008)

	Ref	Expression
Assertion	nebi	⊢ ( ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) ↔ ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )
2	1	necon3bid	⊢ ( ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) → ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )
3		id	⊢ ( ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) → ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )
4	3	necon4bid	⊢ ( ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )
5	2 4	impbii	⊢ ( ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) ↔ ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )