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

Theorem iota5 6051
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 2022 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
3 eqeq2 2776 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
43bibi2d 333 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
54albidv 2015 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
6 eqeq2 2776 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
75, 6imbi12d 335 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
8 iotaval 6042 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
97, 8vtoclg 3418 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
109adantl 473 . 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 197  wa 384  wal 1650   = wceq 1652  wcel 2155  cio 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-v 3352  df-sbc 3597  df-un 3737  df-sn 4335  df-pr 4337  df-uni 4595  df-iota 6031
This theorem is referenced by:  isf32lem9  9436  rlimdm  14569  fsum  14738  fprod  14956  gsumval2a  17547  dchrptlem1  25280  lgsdchrval  25370  iota0def  41747  rlimdmafv  41857  rlimdmafv2  41938
  Copyright terms: Public domain W3C validator