MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon1abid Structured version   Visualization version   GIF version

Theorem necon1abid 2819
Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon1abid.1 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon1abid (𝜑 → (𝐴𝐵𝜓))

Proof of Theorem necon1abid
StepHypRef Expression
1 notnotb 302 . 2 (𝜓 ↔ ¬ ¬ 𝜓)
2 necon1abid.1 . . 3 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
32necon3bbid 2818 . 2 (𝜑 → (¬ ¬ 𝜓𝐴𝐵))
41, 3syl5rbb 271 1 (𝜑 → (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wne 2779
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 2781
This theorem is referenced by:  lttri2  9971  xrlttri2  11810  ioon0  12028  lssne0  18718  xmetgt0  21914
  Copyright terms: Public domain W3C validator