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Theorem clel4 3191
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 2477 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
31, 2ceqsalv 3089 . 2  |-  ( A. x ( x  =  B  ->  A  e.  x )  <->  A  e.  B )
43bicomi 204 1  |-  ( A  e.  B  <->  A. x
( x  =  B  ->  A  e.  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186   A.wal 1405    = wceq 1407    e. wcel 1844   _Vcvv 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-12 1880  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-v 3063
This theorem is referenced by:  intpr  4263
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