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Theorem dfaiota3 45800
Description: Alternate definition of ℩': this is to df-aiota 45793 what dfiota4 6536 is to df-iota 6496. operation using the if operator. It is simpler than df-aiota 45793 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 45799 . 2 (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})
2 aiotavb 45798 . . 3 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
32biimpi 215 . 2 (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4 ifval 4571 . 2 ((℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
51, 3, 4mpbir2an 710 1 (℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  ∃!weu 2563  {cab 2710  Vcvv 3475  ifcif 4529   cint 4951  ℩'caiota 45791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-uni 4910  df-int 4952  df-iota 6496  df-aiota 45793
This theorem is referenced by: (None)
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