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Theorem neleq1 2537
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 2343 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21notbid 285 . 2  |-  ( A  =  B  ->  ( -.  A  e.  C  <->  -.  B  e.  C ) )
3 df-nel 2449 . 2  |-  ( A  e/  C  <->  -.  A  e.  C )
4 df-nel 2449 . 2  |-  ( B  e/  C  <->  -.  B  e.  C )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
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:  ruALT  7315  cnpart  11725  sqrmo  11737  resqrcl  11739  resqrthlem  11740  sqrneg  11753  sqreu  11844  sqrthlem  11846  eqsqrd  11851  iccpnfcnv  18442  xrge0iifcnv  23315  neleq12d  24933  gltpntl  26072  isibg2aa  26112  isibg2aalem1  26113  isibg2aalem2  26114
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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