MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon2abii Structured version   Visualization version   GIF version

Theorem necon2abii 2831
Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
Hypothesis
Ref Expression
necon2abii.1 (𝐴 = 𝐵 ↔ ¬ 𝜑)
Assertion
Ref Expression
necon2abii (𝜑𝐴𝐵)

Proof of Theorem necon2abii
StepHypRef Expression
1 necon2abii.1 . . . 4 (𝐴 = 𝐵 ↔ ¬ 𝜑)
21bicomi 212 . . 3 𝜑𝐴 = 𝐵)
32necon1abii 2829 . 2 (𝐴𝐵𝜑)
43bicomi 212 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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:  locfindis  21085  flimsncls  21542  tsmsgsum  21694  wilthlem2  24512  topdifinffinlem  32167  ismblfin  32416  elnev  37457
  Copyright terms: Public domain W3C validator