Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iota5 Structured version   Visualization version   GIF version

Theorem iota5 6307
 Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)
Hypothesis
Ref Expression
iota5.1 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
Assertion
Ref Expression
iota5 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem iota5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iota5.1 . . 3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
21alrimiv 1928 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
3 eqeq2 2810 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
43bibi2d 346 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
54albidv 1921 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
6 eqeq2 2810 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
75, 6imbi12d 348 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
8 iotaval 6298 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
97, 8vtoclg 3515 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
109adantl 485 . 2 ((𝜑𝐴𝑉) → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
112, 10mpd 15 1 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ℩cio 6281 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283 This theorem is referenced by:  isf32lem9  9772  rlimdm  14900  fsum  15069  fprod  15287  gsumval2a  17887  dchrptlem1  25848  lgsdchrval  25938  iota0def  43625  rlimdmafv  43728  rlimdmafv2  43809
 Copyright terms: Public domain W3C validator