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Theorem neleq2 2385
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 2181 . . 3  |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
21notbid 641 . 2  |-  ( A  =  B  ->  ( -.  C  e.  A  <->  -.  C  e.  B ) )
3 df-nel 2381 . 2  |-  ( C  e/  A  <->  -.  C  e.  A )
4 df-nel 2381 . 2  |-  ( C  e/  B  <->  -.  C  e.  B )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465    e/ wnel 2380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113  df-nel 2381
This theorem is referenced by:  neleq12d  2386
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