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Theorem aiotaexb 46369
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 5332 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
2 euabsn2 4724 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
3 df-aiota 46365 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
43eleq1i 2818 . 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 1533  wex 1773  wcel 2098  ∃!weu 2556  {cab 2703  Vcvv 3468  {csn 4623   cint 4943  ℩'caiota 46363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-int 4944  df-aiota 46365
This theorem is referenced by:  aiotavb  46370  iotan0aiotaex  46373  aiotaexaiotaiota  46374  aiota0ndef  46377
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