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Theorem clel3g 2678
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
Assertion
Ref Expression
clel3g  |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem clel3g
StepHypRef Expression
1 eleq2 2101 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
21ceqsexgv 2673 . 2  |-  ( B  e.  V  ->  ( E. x ( x  =  B  /\  A  e.  x )  <->  A  e.  B ) )
32bicomd 129 1  |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  clel3  2679  dfiun2g  3689
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