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Theorem neleq2 2345
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
Assertion
Ref Expression
neleq2  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )

Proof of Theorem neleq2
StepHypRef Expression
1 eleq2 2143 . . 3  |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
21notbid 625 . 2  |-  ( A  =  B  ->  ( -.  C  e.  A  <->  -.  C  e.  B ) )
3 df-nel 2341 . 2  |-  ( C  e/  A  <->  -.  C  e.  A )
4 df-nel 2341 . 2  |-  ( C  e/  B  <->  -.  C  e.  B )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434    e/ wnel 2340
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 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-nel 2341
This theorem is referenced by:  neleq12d  2346
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