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Theorem intexabim 3994
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 2077 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1542 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfsab1 2079 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1467 . . . 4  |-  F/ y  x  e.  { x  |  ph }
5 eleq1 2151 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
63, 4, 5cbvex 1687 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x  x  e.  {
x  |  ph }
)
7 inteximm 3991 . . 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 1427    e. wcel 1439   {cab 2075   _Vcvv 2620   |^|cint 3694
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-int 3695
This theorem is referenced by:  intexrabim  3995  omex  4421
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