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Theorem necon1bbii 2707
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon1bbii.1  |-  ( A  =/=  B  <->  ph )
Assertion
Ref Expression
necon1bbii  |-  ( -. 
ph 
<->  A  =  B )

Proof of Theorem necon1bbii
StepHypRef Expression
1 nne 2644 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
2 necon1bbii.1 . 2  |-  ( A  =/=  B  <->  ph )
31, 2xchnxbi 308 1  |-  ( -. 
ph 
<->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1383    =/= wne 2638
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 185  df-ne 2640
This theorem is referenced by:  necon2bbii  2710  rabeq0  3793  intnex  4594  class2set  4604  csbopab  4769  relimasn  5350  modom  7722  supval2  7917  fzo0  11828  vma1  23312  lgsquadlem3  23503  ordtconlem1  27779
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