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

Theorem reuaiotaiota 47037
Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
reuaiotaiota (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem reuaiotaiota
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsneu 46977 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
2 reuabaiotaiota 47036 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitri 275 1 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  ∃!weu 2565  {cab 2711  {csn 4630  cio 6513  ℩'caiota 47032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-uni 4912  df-int 4951  df-iota 6515  df-aiota 47034
This theorem is referenced by:  aiotaint  47040  aiotaexaiotaiota  47043  dfafv2  47081
  Copyright terms: Public domain W3C validator