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Theorem neleq2 2538
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 2344 . . 3  |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
21notbid 285 . 2  |-  ( A  =  B  ->  ( -.  C  e.  A  <->  -.  C  e.  B ) )
3 df-nel 2449 . 2  |-  ( C  e/  A  <->  -.  C  e.  A )
4 df-nel 2449 . 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 1623    e. wcel 1684    e/ wnel 2447
This theorem is referenced by:  noinfep  7360  isfbas  17524  neleq12d  24933  isibg2aa  26112  isibg2aalem3  26115  nbgra0nb  28144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279  df-nel 2449
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