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Theorem dfiota4 6425
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 6408 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iotanul 6411 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3 ifval 4501 . 2 ((℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
41, 2, 3mpbir2an 708 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  ∃!weu 2568  {cab 2715  c0 4256  ifcif 4459   cuni 4839  cio 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391
This theorem is referenced by: (None)
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