Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iota5f Structured version   Visualization version   GIF version

Theorem iota5f 32852
Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
Hypotheses
Ref Expression
iota5f.1 𝑥𝜑
iota5f.2 𝑥𝐴
iota5f.3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
Assertion
Ref Expression
iota5f ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem iota5f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iota5f.1 . . . 4 𝑥𝜑
2 iota5f.2 . . . . 5 𝑥𝐴
32nfel1 2991 . . . 4 𝑥 𝐴𝑉
41, 3nfan 1891 . . 3 𝑥(𝜑𝐴𝑉)
5 iota5f.3 . . 3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
64, 5alrimi 2203 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
72nfeq2 2992 . . . . . 6 𝑥 𝑦 = 𝐴
8 eqeq2 2830 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
98bibi2d 344 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
107, 9albid 2214 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
11 eqeq2 2830 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
1210, 11imbi12d 346 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13 iotaval 6322 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
1412, 13vtoclg 3565 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
1514adantl 482 . 2 ((𝜑𝐴𝑉) → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
166, 15mpd 15 1 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958  cio 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator