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Theorem intexabim 4009
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 2083 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1548 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfsab1 2085 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1473 . . . 4  |-  F/ y  x  e.  { x  |  ph }
5 eleq1 2157 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
63, 4, 5cbvex 1693 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x  x  e.  {
x  |  ph }
)
7 inteximm 4006 . . 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 1433    e. wcel 1445   {cab 2081   _Vcvv 2633   |^|cint 3710
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-int 3711
This theorem is referenced by:  intexrabim  4010  omex  4436
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