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Theorem iotauni 6401
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 eu6 2574 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 iotaval 6400 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3 uniabio 6399 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → {𝑥𝜑} = 𝑧)
42, 3eqtr4d 2781 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
54exlimiv 1933 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = {𝑥𝜑})
61, 5sylbi 216 1 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wex 1782  ∃!weu 2568  {cab 2715   cuni 4839  cio 6382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3431  df-un 3891  df-in 3893  df-ss 3903  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6384
This theorem is referenced by:  iotaint  6402  iotassuni  6405  dfiota4  6418  fveu  6755  riotauni  7230  afv2eu  44708
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