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Theorem elint 3868
Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)
Hypothesis
Ref Expression
elint.1  |-  A  e. 
_V
Assertion
Ref Expression
elint  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem elint
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elint.1 . 2  |-  A  e. 
_V
2 eleq1 2343 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 307 . . 3  |-  ( y  =  A  ->  (
( x  e.  B  ->  y  e.  x )  <-> 
( x  e.  B  ->  A  e.  x ) ) )
43albidv 1611 . 2  |-  ( y  =  A  ->  ( A. x ( x  e.  B  ->  y  e.  x )  <->  A. x
( x  e.  B  ->  A  e.  x ) ) )
5 df-int 3863 . 2  |-  |^| B  =  { y  |  A. x ( x  e.  B  ->  y  e.  x ) }
61, 4, 5elab2 2917 1  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   |^|cint 3862
This theorem is referenced by:  elint2  3869  elintab  3873  intss1  3877  intss  3883  intun  3894  intpr  3895  cssmre  16593  dfom5b  24452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-int 3863
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