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Theorem aiotaexb 47101
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 5346 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
2 euabsn2 4725 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
3 df-aiota 47097 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
43eleq1i 2832 . 2 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
51, 2, 43bitr4i 303 1 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  wcel 2108  ∃!weu 2568  {cab 2714  Vcvv 3480  {csn 4626   cint 4946  ℩'caiota 47095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-int 4947  df-aiota 47097
This theorem is referenced by:  aiotavb  47102  iotan0aiotaex  47105  aiotaexaiotaiota  47106  aiota0ndef  47109
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