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Theorem elintg 4290
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 2539 . 2  |-  ( y  =  A  ->  (
y  e.  |^| B  <->  A  e.  |^| B ) )
2 eleq1 2539 . . 3  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32ralbidv 2903 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  e.  x ) )
4 vex 3116 . . 3  |-  y  e. 
_V
54elint2 4289 . 2  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
61, 3, 5vtoclbg 3172 1  |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2814   |^|cint 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-int 4283
This theorem is referenced by:  elinti  4291  elrint  4323  onmindif  4967  onmindif2  6625  mremre  14855  toponmre  19360  1stcfb  19712  uffixfr  20159  plycpn  22419  insiga  27777  dfon2lem8  28799  trintALTVD  32760  trintALT  32761
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