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Theorem elint 3814
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 2220 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 229 . . 3  |-  ( y  =  A  ->  (
( x  e.  B  ->  y  e.  x )  <-> 
( x  e.  B  ->  A  e.  x ) ) )
43albidv 1804 . 2  |-  ( y  =  A  ->  ( A. x ( x  e.  B  ->  y  e.  x )  <->  A. x
( x  e.  B  ->  A  e.  x ) ) )
5 df-int 3809 . 2  |-  |^| B  =  { y  |  A. x ( x  e.  B  ->  y  e.  x ) }
61, 4, 5elab2 2860 1  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1333    = wceq 1335    e. wcel 2128   _Vcvv 2712   |^|cint 3808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-int 3809
This theorem is referenced by:  elint2  3815  elintab  3819  intss1  3823  intss  3829  intun  3839  intpr  3840  peano1  4554
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