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Theorem dfaiota3 46519
Description: Alternate definition of ℩': this is to df-aiota 46512 what dfiota4 6545 is to df-iota 6505. operation using the if operator. It is simpler than df-aiota 46512 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 46518 . 2 (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})
2 aiotavb 46517 . . 3 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
32biimpi 215 . 2 (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4 ifval 4574 . 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 1533  ∃!weu 2557  {cab 2705  Vcvv 3473  ifcif 4532   cint 4953  ℩'caiota 46510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-uni 4913  df-int 4954  df-iota 6505  df-aiota 46512
This theorem is referenced by: (None)
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