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Theorem neleq2 2551
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 2357 . . 3  |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
21notbid 285 . 2  |-  ( A  =  B  ->  ( -.  C  e.  A  <->  -.  C  e.  B ) )
3 df-nel 2462 . 2  |-  ( C  e/  A  <->  -.  C  e.  A )
4 df-nel 2462 . 2  |-  ( C  e/  B  <->  -.  C  e.  B )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    e/ wnel 2460
This theorem is referenced by:  noinfep  7376  isfbas  17540  neleq12d  25036  isibg2aa  26215  isibg2aalem3  26218  nbgra0nb  28278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292  df-nel 2462
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