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Theorem clel4 2848
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 2221 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
31, 2ceqsalv 2742 . 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 1333    = wceq 1335    e. wcel 2128   _Vcvv 2712
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  intpr  3839
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