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Theorem intexabim 4136
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 2158 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1598 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfsab1 2160 . . . 4  |-  F/ x  y  e.  { x  |  ph }
4 nfv 1521 . . . 4  |-  F/ y  x  e.  { x  |  ph }
5 eleq1 2233 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  x  e.  { x  |  ph }
) )
63, 4, 5cbvex 1749 . . 3  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x  x  e.  {
x  |  ph }
)
7 inteximm 4133 . . 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 1485    e. wcel 2141   {cab 2156   _Vcvv 2730   |^|cint 3829
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4105
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3830
This theorem is referenced by:  intexrabim  4137  omex  4575
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