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Theorem inex1g 4024
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 3236 . . 3  |-  ( x  =  A  ->  (
x  i^i  B )  =  ( A  i^i  B ) )
21eleq1d 2183 . 2  |-  ( x  =  A  ->  (
( x  i^i  B
)  e.  _V  <->  ( A  i^i  B )  e.  _V ) )
3 vex 2660 . . 3  |-  x  e. 
_V
43inex1 4022 . 2  |-  ( x  i^i  B )  e. 
_V
52, 4vtoclg 2717 1  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2657    i^i cin 3036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043
This theorem is referenced by:  onin  4268  dmresexg  4800  funimaexg  5165  offval  5943  offval3  5986  ssenen  6698  ressval2  11862  eltg  12064  eltg3  12069  ntrval  12122  restco  12186
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