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Theorem iotauni 4930
 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 1946 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 iotaval 4929 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3 uniabio 4928 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → {𝑥𝜑} = 𝑧)
42, 3eqtr4d 2118 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
54exlimiv 1530 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
61, 5sylbi 119 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1283   = wceq 1285  ∃wex 1422  ∃!weu 1943  {cab 2069  ∪ cuni 3622  ℩cio 4916 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-sbc 2826  df-un 2987  df-sn 3423  df-pr 3424  df-uni 3623  df-iota 4918 This theorem is referenced by:  iotaint  4931  fveu  5222  riotauni  5526
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