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Theorem neleq1 2348
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
Assertion
Ref Expression
neleq1  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )

Proof of Theorem neleq1
StepHypRef Expression
1 eleq1 2145 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21notbid 625 . 2  |-  ( A  =  B  ->  ( -.  A  e.  C  <->  -.  B  e.  C ) )
3 df-nel 2345 . 2  |-  ( A  e/  C  <->  -.  A  e.  C )
4 df-nel 2345 . 2  |-  ( B  e/  C  <->  -.  B  e.  C )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434    e/ wnel 2344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-nel 2345
This theorem is referenced by:  neleq12d  2350  ruALT  4323
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