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Theorem necon1bbii 2988
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 2942 . 2 𝐴𝐵𝐴 = 𝐵)
2 necon1bbii.1 . 2 (𝐴𝐵𝜑)
31, 2xchnxbi 332 1 𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wne 2938
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 207  df-ne 2939
This theorem is referenced by:  necon2bbii  2990  intnex  5351  class2set  5361  csbopab  5565  relimasn  6105  modom  9278  supval2  9493  fzo0  13720  vma1  27224  lgsquadlem3  27441  ordtconnlem1  33885
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