Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfaiota3 Structured version   Visualization version   GIF version

Theorem dfaiota3 47104
Description: Alternate definition of ℩': this is to df-aiota 47097 what dfiota4 6553 is to df-iota 6514. operation using the if operator. It is simpler than df-aiota 47097 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 47103 . 2 (∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑})
2 aiotavb 47102 . . 3 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
32biimpi 216 . 2 (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4 ifval 4568 . 2 ((℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = {𝑥𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
51, 3, 4mpbir2an 711 1 (℩'𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  ∃!weu 2568  {cab 2714  Vcvv 3480  ifcif 4525   cint 4946  ℩'caiota 47095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-iota 6514  df-aiota 47097
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator