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Theorem aiotaexb 44581
Description: The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiotaexb (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Proof of Theorem aiotaexb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 intexab 5263 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
2 euabsn2 4661 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
3 df-aiota 44577 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
43eleq1i 2829 . 2 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
51, 2, 43bitr4i 303 1 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wex 1782  wcel 2106  ∃!weu 2568  {cab 2715  Vcvv 3432  {csn 4561   cint 4879  ℩'caiota 44575
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  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  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-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-int 4880  df-aiota 44577
This theorem is referenced by:  aiotavb  44582  iotan0aiotaex  44585  aiotaexaiotaiota  44586  aiota0ndef  44589
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