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

Theorem aiotaexaiotaiota 44039
 Description: The alternate iota over a wff 𝜑 is a set iff the iota and the alternate iota over 𝜑 are equal. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiotaexaiotaiota ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem aiotaexaiotaiota
StepHypRef Expression
1 aiotaexb 44034 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
2 reuaiotaiota 44033 . 2 (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
31, 2bitr3i 280 1 ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃!weu 2587  Vcvv 3409  ℩cio 6292  ℩'caiota 44028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-pr 4525  df-uni 4799  df-int 4839  df-iota 6294  df-aiota 44030 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator