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Theorem reuaiotaiota 44638
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 44580 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
2 reuabaiotaiota 44637 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitri 275 1 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  ∃!weu 2566  {cab 2713  {csn 4565  cio 6408  ℩'caiota 44633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-sn 4566  df-pr 4568  df-uni 4845  df-int 4887  df-iota 6410  df-aiota 44635
This theorem is referenced by:  aiotaint  44641  aiotaexaiotaiota  44644  dfafv2  44682
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