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Theorem intexabim 4173
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 2177 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1616 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfsab1 2179 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1539 . . . 4  |-  F/ y  x  e.  { x  |  ph }
5 eleq1 2252 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
63, 4, 5cbvex 1767 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x  x  e.  {
x  |  ph }
)
7 inteximm 4170 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  ->  |^| { x  | 
ph }  e.  _V )
86, 7sylbir 135 . 2  |-  ( E. x  x  e.  {
x  |  ph }  ->  |^| { x  | 
ph }  e.  _V )
92, 8sylbir 135 1  |-  ( E. x ph  ->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1503    e. wcel 2160   {cab 2175   _Vcvv 2752   |^|cint 3862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4139
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-int 3863
This theorem is referenced by:  intexrabim  4174  omex  4613
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