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Theorem aiotavb 43280
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 5232 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
2 df-aiota 43275 . . . . 5 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
32eleq1i 2901 . . . 4 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
43notbii 322 . . 3 (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
52eqeq1i 2824 . . 3 ((℩'𝑥𝜑) = V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
61, 4, 53bitr4i 305 . 2 (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7 aiotaexb 43279 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
86, 7xchnxbir 335 1 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208   = wceq 1530  wcel 2107  ∃!weu 2647  {cab 2797  Vcvv 3493  {csn 4559   cint 4867  ℩'caiota 43273
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rab 3145  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-int 4868  df-aiota 43275
This theorem is referenced by:  dfafv2  43321
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