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Theorem dfiota4 6325
Description: The operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.)
Assertion
Ref Expression
dfiota4 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Proof of Theorem dfiota4
StepHypRef Expression
1 iotauni 6308 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iotanul 6311 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3 ifval 4453 . 2 ((℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
41, 2, 3mpbir2an 711 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  ∃!weu 2569  {cab 2716  c0 4209  ifcif 4411   cuni 4793  cio 6289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-uni 4794  df-iota 6291
This theorem is referenced by: (None)
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