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

Theorem reuabaiotaiota 45730
Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
reuabaiotaiota (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuabaiotaiota
StepHypRef Expression
1 uniintab 4991 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
2 df-iota 6492 . . 3 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
3 df-aiota 45728 . . 3 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
42, 3eqeq12i 2751 . 2 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ {𝑥𝜑} = {𝑦}})
51, 4bitr4i 278 1 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  ∃!weu 2563  {cab 2710  {csn 4627   cuni 4907   cint 4949  cio 6490  ℩'caiota 45726
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
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-int 4950  df-iota 6492  df-aiota 45728
This theorem is referenced by:  reuaiotaiota  45731  aiotaval  45738
  Copyright terms: Public domain W3C validator