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Theorem elintg 3886
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 2356 . 2  |-  ( y  =  A  ->  (
y  e.  |^| B  <->  A  e.  |^| B ) )
2 eleq1 2356 . . 3  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32ralbidv 2576 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  e.  x ) )
4 vex 2804 . . 3  |-  y  e. 
_V
54elint2 3885 . 2  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
61, 3, 5vtoclbg 2857 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 176    = wceq 1632    e. wcel 1696   A.wral 2556   |^|cint 3878
This theorem is referenced by:  elinti  3887  elrint  3919  onmindif  4498  onmindif2  4619  mremre  13522  toponmre  16846  1stcfb  17187  uffixfr  17634  plycpn  19685  insiga  23513  dfon2lem8  24217  intfmu2  25622  prcnt  25654  trintALTVD  28972  trintALT  28973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-int 3879
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