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Theorem intexabim 4047
Description: The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexabim  |-  ( E. x ph  ->  |^| { x  |  ph }  e.  _V )

Proof of Theorem intexabim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 abid 2105 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1569 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfsab1 2107 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1493 . . . 4  |-  F/ y  x  e.  { x  |  ph }
5 eleq1 2180 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
63, 4, 5cbvex 1714 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x  x  e.  {
x  |  ph }
)
7 inteximm 4044 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  ->  |^| { x  | 
ph }  e.  _V )
86, 7sylbir 134 . 2  |-  ( E. x  x  e.  {
x  |  ph }  ->  |^| { x  | 
ph }  e.  _V )
92, 8sylbir 134 1  |-  ( E. x ph  ->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1453    e. wcel 1465   {cab 2103   _Vcvv 2660   |^|cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by:  intexrabim  4048  omex  4477
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