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Theorem necon1bbii 2981
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 2936 . 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 1632  wne 2932
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 2933
This theorem is referenced by:  necon2bbii  2983  rabeq0OLD  4103  intnex  4970  class2set  4981  csbopab  5158  relimasn  5646  modom  8326  supval2  8526  fzo0  12686  vma1  25091  lgsquadlem3  25306  ordtconnlem1  30279
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