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Theorem uniintabim 3725
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 3511 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsnr 3724 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑥𝜑})
31, 2sylbi 119 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wex 1426  ∃!weu 1948  {cab 2074  {csn 3446   cuni 3653   cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689
This theorem is referenced by:  iotaint  4993
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