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Theorem elint 3694
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 2150 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 228 . . 3  |-  ( y  =  A  ->  (
( x  e.  B  ->  y  e.  x )  <-> 
( x  e.  B  ->  A  e.  x ) ) )
43albidv 1752 . 2  |-  ( y  =  A  ->  ( A. x ( x  e.  B  ->  y  e.  x )  <->  A. x
( x  e.  B  ->  A  e.  x ) ) )
5 df-int 3689 . 2  |-  |^| B  =  { y  |  A. x ( x  e.  B  ->  y  e.  x ) }
61, 4, 5elab2 2763 1  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   |^|cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-int 3689
This theorem is referenced by:  elint2  3695  elintab  3699  intss1  3703  intss  3709  intun  3719  intpr  3720  peano1  4409
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