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Theorem clel3g 2918
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 2357 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
21ceqsexgv 2913 . 2  |-  ( B  e.  V  ->  ( E. x ( x  =  B  /\  A  e.  x )  <->  A  e.  B ) )
32bicomd 192 1  |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  clel3  2919  dfiun2g  3951
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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