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Theorem elint2 3650
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
elint2.1  |-  A  e. 
_V
Assertion
Ref Expression
elint2  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Distinct variable groups:    x, A    x, B

Proof of Theorem elint2
StepHypRef Expression
1 elint2.1 . . 3  |-  A  e. 
_V
21elint 3649 . 2  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
3 df-ral 2328 . 2  |-  ( A. x  e.  B  A  e.  x  <->  A. x ( x  e.  B  ->  A  e.  x ) )
42, 3bitr4i 180 1  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    e. wcel 1409   A.wral 2323   _Vcvv 2574   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-int 3644
This theorem is referenced by:  elintg  3651  ssint  3659  intssunim  3665  iinuniss  3765  trint  3897  trintssmOLD  3899
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