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Theorem clel3g 3075
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 2499 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
21ceqsexgv 3070 . 2  |-  ( B  e.  V  ->  ( E. x ( x  =  B  /\  A  e.  x )  <->  A  e.  B ) )
32bicomd 194 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726
This theorem is referenced by:  clel3  3076  dfiun2g  4125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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