Theorem necon3bii

Description: Inference from equality to inequality. (Contributed by NM, 23-Feb-2005)

	Ref	Expression
Hypothesis	necon3bii.1	⊢ ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 )
Assertion	necon3bii	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		necon3bii.1	⊢ ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 )
2	1	necon3abii	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷 )
3		df-ne	⊢ ( 𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷 )
4	2 3	bitr4i	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 )