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Theorem nebi 2857
Description: Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
Assertion
Ref Expression
nebi ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))

Proof of Theorem nebi
StepHypRef Expression
1 id 22 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐵𝐶 = 𝐷))
21necon3bid 2821 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐵𝐶𝐷))
3 id 22 . . 3 ((𝐴𝐵𝐶𝐷) → (𝐴𝐵𝐶𝐷))
43necon4bid 2822 . 2 ((𝐴𝐵𝐶𝐷) → (𝐴 = 𝐵𝐶 = 𝐷))
52, 4impbii 197 1 ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wne 2775
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 195  df-ne 2777
This theorem is referenced by: (None)
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