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Theorem elintrab 3695
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
Hypothesis
Ref Expression
inteqab.1  |-  A  e. 
_V
Assertion
Ref Expression
elintrab  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elintrab
StepHypRef Expression
1 inteqab.1 . . . 4  |-  A  e. 
_V
21elintab 3694 . . 3  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  e.  x ) )
3 impexp 259 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  e.  x )  <->  ( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
43albii 1404 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
52, 4bitri 182 . 2  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
6 df-rab 2368 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
76inteqi 3687 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
87eleq2i 2154 . 2  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  |  ( x  e.  B  /\  ph ) } )
9 df-ral 2364 . 2  |-  ( A. x  e.  B  ( ph  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
105, 8, 93bitr4i 210 1  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    e. wcel 1438   {cab 2074   A.wral 2359   {crab 2363   _Vcvv 2619   |^|cint 3683
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-ral 2364  df-rab 2368  df-v 2621  df-int 3684
This theorem is referenced by:  elintrabg  3696  intmin  3703  bj-indint  11472
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