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Theorem necon1bbii 3036
 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 2991 . 2 𝐴𝐵𝐴 = 𝐵)
2 necon1bbii.1 . 2 (𝐴𝐵𝜑)
31, 2xchnxbi 335 1 𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ≠ wne 2987 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 210  df-ne 2988 This theorem is referenced by:  necon2bbii  3038  intnex  5209  class2set  5223  csbopab  5411  relimasn  5923  modom  8721  supval2  8921  fzo0  13076  vma1  25795  lgsquadlem3  26010  ordtconnlem1  31343
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