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Theorem clel2 3064
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 2495 . . 3  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
31, 2ceqsalv 2974 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  A  e.  B )
43bicomi 194 1  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem is referenced by:  snss  3918  mptelee  25799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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