Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon1bbii Structured version   Visualization version   GIF version

Theorem necon1bbii 2973
 Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon1bbii.1 (𝐴𝐵𝜑)
Assertion
Ref Expression
necon1bbii 𝜑𝐴 = 𝐵)

Proof of Theorem necon1bbii
StepHypRef Expression
1 nne 2928 . 2 𝐴𝐵𝐴 = 𝐵)
2 necon1bbii.1 . 2 (𝐴𝐵𝜑)
31, 2xchnxbi 321 1 𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1624   ≠ wne 2924 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 2925 This theorem is referenced by:  necon2bbii  2975  rabeq0OLD  4095  intnex  4962  class2set  4973  csbopab  5150  relimasn  5638  modom  8318  supval2  8518  fzo0  12678  vma1  25083  lgsquadlem3  25298  ordtconnlem1  30271
 Copyright terms: Public domain W3C validator