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Theorem clel4 2866
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel4.1  |-  B  e. 
_V
Assertion
Ref Expression
clel4  |-  ( A  e.  B  <->  A. x
( x  =  B  ->  A  e.  x
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3  |-  B  e. 
_V
2 eleq2 2234 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
31, 2ceqsalv 2760 . 2  |-  ( A. x ( x  =  B  ->  A  e.  x )  <->  A  e.  B )
43bicomi 131 1  |-  ( A  e.  B  <->  A. x
( x  =  B  ->  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346    = wceq 1348    e. wcel 2141   _Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by:  intpr  3863
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