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Theorem reuaiotaiota 46530
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 46472 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
2 reuabaiotaiota 46529 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitri 274 1 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  ∃!weu 2556  {cab 2702  {csn 4624  cio 6492  ℩'caiota 46525
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-sn 4625  df-pr 4627  df-uni 4904  df-int 4945  df-iota 6494  df-aiota 46527
This theorem is referenced by:  aiotaint  46533  aiotaexaiotaiota  46536  dfafv2  46574
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