Theorem nebi 3006

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

Assertion

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

Proof of Theorem nebi

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
2	1	necon3bid 2970	. 2 ⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
3		id 22	. . 3 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
4	3	necon4bid 2971	. 2 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
5	2, 4	impbii 209	1 ⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))

Syntax hints:   ↔ wb 206   = wceq 1540   ≠ wne 2926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-ne 2927

This theorem is referenced by: (None)