Theorem neanior

Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007)

	Ref	Expression
Assertion	neanior	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ) ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
2		df-ne	⊢ ( 𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷 )
3	1 2	anbi12i	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ) ↔ ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ) )
4		pm4.56	⊢ ( ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ) ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ) )
5	3 4	bitri	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ) ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ) )