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Theorem elint 3884
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 2356 . . . 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 1615 . 2  |-  ( y  =  A  ->  ( A. x ( x  e.  B  ->  y  e.  x )  <->  A. x
( x  e.  B  ->  A  e.  x ) ) )
5 df-int 3879 . 2  |-  |^| B  =  { y  |  A. x ( x  e.  B  ->  y  e.  x ) }
61, 4, 5elab2 2930 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 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801   |^|cint 3878
This theorem is referenced by:  elint2  3885  elintab  3889  intss1  3893  intss  3899  intun  3910  intpr  3911  cssmre  16609  dfom5b  24523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-int 3879
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