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Theorem iotauni 5070
Description: Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iotauni (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Proof of Theorem iotauni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 1980 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 iotaval 5069 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3 uniabio 5068 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → {𝑥𝜑} = 𝑧)
42, 3eqtr4d 2153 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
54exlimiv 1562 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
61, 5sylbi 120 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314   = wceq 1316  wex 1453  ∃!weu 1977  {cab 2103   cuni 3706  cio 5056
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707  df-iota 5058
This theorem is referenced by:  iotaint  5071  fveu  5381  riotauni  5704
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