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Theorem dfiota4 5796
 Description: The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
dfiota4 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Proof of Theorem dfiota4
StepHypRef Expression
1 iotauni 5780 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iftrue 4042 . . 3 (∃!𝑥𝜑 → if(∃!𝑥𝜑, {𝑥𝜑}, ∅) = {𝑥𝜑})
31, 2eqtr4d 2647 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅))
4 iotanul 5783 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 iffalse 4045 . . 3 (¬ ∃!𝑥𝜑 → if(∃!𝑥𝜑, {𝑥𝜑}, ∅) = ∅)
64, 5eqtr4d 2647 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅))
73, 6pm2.61i 175 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475  ∃!weu 2458  {cab 2596  ∅c0 3874  ifcif 4036  ∪ cuni 4372  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768 This theorem is referenced by: (None)
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