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Theorem iotauni 4822
 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 1900 . 2 (∃!xφzx(φx = z))
2 iotaval 4821 . . . 4 (x(φx = z) → (℩xφ) = z)
3 uniabio 4820 . . . 4 (x(φx = z) → {xφ} = z)
42, 3eqtr4d 2072 . . 3 (x(φx = z) → (℩xφ) = {xφ})
54exlimiv 1486 . 2 (zx(φx = z) → (℩xφ) = {xφ})
61, 5sylbi 114 1 (∃!xφ → (℩xφ) = {xφ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378  ∃!weu 1897  {cab 2023  ∪ cuni 3571  ℩cio 4808 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810 This theorem is referenced by:  iotaint  4823  fveu  5113  riotauni  5417
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