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Theorem necon1bbii 3065
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 3020 . 2 𝐴𝐵𝐴 = 𝐵)
2 necon1bbii.1 . 2 (𝐴𝐵𝜑)
31, 2xchnxbi 333 1 𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wne 3016
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 208  df-ne 3017
This theorem is referenced by:  necon2bbii  3067  intnex  5233  class2set  5246  csbopab  5434  relimasn  5946  modom  8708  supval2  8908  fzo0  13051  vma1  25671  lgsquadlem3  25886  ordtconnlem1  31067
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