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Theorem uniintabim 3868
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 3652 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsnr 3867 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑥𝜑})
31, 2sylbi 120 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wex 1485  ∃!weu 2019  {cab 2156  {csn 3583   cuni 3796   cint 3831
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
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832
This theorem is referenced by:  iotaint  5173
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