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Theorem reuaiotaiota 46381
Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
reuaiotaiota (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem reuaiotaiota
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsneu 46323 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
2 reuabaiotaiota 46380 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitri 275 1 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  ∃!weu 2557  {cab 2704  {csn 4624  cio 6492  ℩'caiota 46376
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 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4625  df-pr 4627  df-uni 4904  df-int 4945  df-iota 6494  df-aiota 46378
This theorem is referenced by:  aiotaint  46384  aiotaexaiotaiota  46387  dfafv2  46425
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