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Theorem elintg 3623
Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
elintg  |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem elintg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2100 . 2  |-  ( y  =  A  ->  (
y  e.  |^| B  <->  A  e.  |^| B ) )
2 eleq1 2100 . . 3  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32ralbidv 2326 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  e.  x ) )
4 vex 2560 . . 3  |-  y  e. 
_V
54elint2 3622 . 2  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
61, 3, 5vtoclbg 2614 1  |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-int 3616
This theorem is referenced by:  elinti  3624  elrint  3655  peano2  4318  pitonn  6924  peano1nnnn  6928  peano2nnnn  6929  1nn  7925  peano2nn  7926
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