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Theorem dfiota4 6059
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 6043 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iotanul 6046 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3 ifval 4284 . 2 ((℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
41, 2, 3mpbir2an 702 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  ∃!weu 2581  {cab 2751  c0 4079  ifcif 4243   cuni 4594  cio 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-uni 4595  df-iota 6031
This theorem is referenced by: (None)
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