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Theorem neleq1 2805
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
neleq1  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )

Proof of Theorem neleq1
StepHypRef Expression
1 id 22 . 2  |-  ( A  =  B  ->  A  =  B )
2 eqidd 2468 . 2  |-  ( A  =  B  ->  C  =  C )
31, 2neleq12d 2804 1  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e/ wnel 2663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-cleq 2459  df-clel 2462  df-nel 2665
This theorem is referenced by:  neleq12dOLD  2809  ruALT  8028  ssnn0fi  12062  cnpart  13036  sqrmo  13048  resqrtcl  13050  resqrtthlem  13051  sqrtneg  13064  sqreu  13156  sqrtthlem  13158  eqsqrtd  13163  iccpnfcnv  21207  frgrawopreglem4  24752  xrge0iifcnv  27579
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