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Theorem dfaiota3 45879
Description: Alternate definition of ℩': this is to df-aiota 45872 what dfiota4 6535 is to df-iota 6495. operation using the if operator. It is simpler than df-aiota 45872 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)
Assertion
Ref Expression
dfaiota3 (℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V)

Proof of Theorem dfaiota3
StepHypRef Expression
1 aiotaint 45878 . 2 (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})
2 aiotavb 45877 . . 3 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
32biimpi 215 . 2 (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4 ifval 4570 . 2 ((℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
51, 3, 4mpbir2an 709 1 (℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  ∃!weu 2562  {cab 2709  Vcvv 3474  ifcif 4528   cint 4950  ℩'caiota 45870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-uni 4909  df-int 4951  df-iota 6495  df-aiota 45872
This theorem is referenced by: (None)
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