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Theorem dfiota4OLD 5918
 Description: Obsolete proof of dfiota4 5917 as of 28-Oct-2021. (Contributed by Scott Fenton, 6-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dfiota4OLD (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Proof of Theorem dfiota4OLD
StepHypRef Expression
1 iotauni 5901 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 iftrue 4125 . . 3 (∃!𝑥𝜑 → if(∃!𝑥𝜑, {𝑥𝜑}, ∅) = {𝑥𝜑})
31, 2eqtr4d 2688 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅))
4 iotanul 5904 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 iffalse 4128 . . 3 (¬ ∃!𝑥𝜑 → if(∃!𝑥𝜑, {𝑥𝜑}, ∅) = ∅)
64, 5eqtr4d 2688 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅))
73, 6pm2.61i 176 1 (℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1523  ∃!weu 2498  {cab 2637  ∅c0 3948  ifcif 4119  ∪ cuni 4468  ℩cio 5887 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889 This theorem is referenced by: (None)
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