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Theorem necon1bbii 2990
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 2944 . 2 𝐴𝐵𝐴 = 𝐵)
2 necon1bbii.1 . 2 (𝐴𝐵𝜑)
31, 2xchnxbi 331 1 𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1540  wne 2940
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 206  df-ne 2941
This theorem is referenced by:  necon2bbii  2992  intnex  5282  class2set  5296  csbopab  5499  relimasn  6022  modom  9109  supval2  9312  fzo0  13512  vma1  26421  lgsquadlem3  26636  ordtconnlem1  32172
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