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Theorem uniintabim 3652
Description: The union and the intersection of a class abstraction are equal if there is a unique satisfying value of  ph ( x ). (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
uniintabim  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )

Proof of Theorem uniintabim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3439 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsnr 3651 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
31, 2sylbi 114 1  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   E.wex 1381   E!weu 1900   {cab 2026   {csn 3375   U.cuni 3580   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616
This theorem is referenced by:  iotaint  4880
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