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Theorem inex1g 3967
Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
inex1g  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )

Proof of Theorem inex1g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ineq1 3192 . . 3  |-  ( x  =  A  ->  (
x  i^i  B )  =  ( A  i^i  B ) )
21eleq1d 2156 . 2  |-  ( x  =  A  ->  (
( x  i^i  B
)  e.  _V  <->  ( A  i^i  B )  e.  _V ) )
3 vex 2622 . . 3  |-  x  e. 
_V
43inex1 3965 . 2  |-  ( x  i^i  B )  e. 
_V
52, 4vtoclg 2679 1  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619    i^i cin 2996
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  ax-sep 3949
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-v 2621  df-in 3003
This theorem is referenced by:  onin  4204  dmresexg  4723  funimaexg  5084  offval  5845  offval3  5887  ssenen  6547
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