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Theorem aiotavb 44549
Description: The alternate iota over a wff 𝜑 is the universe iff there is no unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiotavb (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)

Proof of Theorem aiotavb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 intnex 5264 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
2 df-aiota 44544 . . . . 5 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
32eleq1i 2829 . . . 4 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
43notbii 320 . . 3 (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
52eqeq1i 2743 . . 3 ((℩'𝑥𝜑) = V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
61, 4, 53bitr4i 303 . 2 (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7 aiotaexb 44548 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
86, 7xchnxbir 333 1 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1539  wcel 2106  ∃!weu 2568  {cab 2715  Vcvv 3431  {csn 4563   cint 4881  ℩'caiota 44542
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 5225
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 3433  df-dif 3891  df-in 3895  df-ss 3905  df-nul 4259  df-sn 4564  df-int 4882  df-aiota 44544
This theorem is referenced by:  dfaiota3  44551  dfafv2  44591
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