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Theorem uniintabim 3844
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 3628 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsnr 3843 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑥𝜑})
31, 2sylbi 120 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wex 1472  ∃!weu 2006  {cab 2143  {csn 3560   cuni 3772   cint 3807
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808
This theorem is referenced by:  iotaint  5148
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