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Theorem uniintabim 3643
 Description: The union and the intersection of a class abstraction are equal if there is a unique satisfying value of φ(x). (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
uniintabim (∃!xφ {xφ} = {xφ})

Proof of Theorem uniintabim
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3430 . 2 (∃!xφy{xφ} = {y})
2 uniintsnr 3642 . 2 (y{xφ} = {y} → {xφ} = {xφ})
31, 2sylbi 114 1 (∃!xφ {xφ} = {xφ})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378  ∃!weu 1897  {cab 2023  {csn 3367  ∪ cuni 3571  ∩ cint 3606 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607 This theorem is referenced by:  iotaint  4823
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