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Theorem clel2 2792
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel2.1  |-  A  e. 
_V
Assertion
Ref Expression
clel2  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . . 3  |-  A  e. 
_V
2 eleq1 2180 . . 3  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
31, 2ceqsalv 2690 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  A  e.  B )
43bicomi 131 1  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   _Vcvv 2660
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  snss  3619
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