MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon4bid Structured version   Visualization version   GIF version

Theorem necon4bid 2836
Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
Hypothesis
Ref Expression
necon4bid.1 (𝜑 → (𝐴𝐵𝐶𝐷))
Assertion
Ref Expression
necon4bid (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Proof of Theorem necon4bid
StepHypRef Expression
1 necon4bid.1 . . 3 (𝜑 → (𝐴𝐵𝐶𝐷))
21necon2bbid 2834 . 2 (𝜑 → (𝐶 = 𝐷 ↔ ¬ 𝐴𝐵))
3 nne 2795 . 2 𝐴𝐵𝐴 = 𝐵)
42, 3syl6rbb 277 1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1481  wne 2791
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 197  df-ne 2792
This theorem is referenced by:  nebi  2871  znnenlem  14921  rpexp  15413  norm-i  27956  trlid0b  35284
  Copyright terms: Public domain W3C validator