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Theorem clel2 3163
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 2537 . . 3  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
31, 2ceqsalv 3061 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  A  e.  B )
43bicomi 207 1  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452    e. wcel 1904   _Vcvv 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033
This theorem is referenced by:  snss  4087  mptelee  25004
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