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Theorem aiotavb 46736
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 5335 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
2 df-aiota 46731 . . . . 5 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
32eleq1i 2817 . . . 4 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
43notbii 319 . . 3 (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
52eqeq1i 2731 . . 3 ((℩'𝑥𝜑) = V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
61, 4, 53bitr4i 302 . 2 (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7 aiotaexb 46735 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
86, 7xchnxbir 332 1 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wcel 2099  ∃!weu 2557  {cab 2703  Vcvv 3462  {csn 4623   cint 4946  ℩'caiota 46729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-in 3953  df-ss 3963  df-nul 4323  df-sn 4624  df-int 4947  df-aiota 46731
This theorem is referenced by:  dfaiota3  46738  dfafv2  46778
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