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Theorem dfiota4 6349
 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 6332 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iotanul 6335 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3 ifval 4510 . 2 ((℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
41, 2, 3mpbir2an 709 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537  ∃!weu 2653  {cab 2801  ∅c0 4293  ifcif 4469  ∪ cuni 4840  ℩cio 6314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-uni 4841  df-iota 6316 This theorem is referenced by: (None)
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