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

Theorem riotav 7317
Description: An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
Assertion
Ref Expression
riotav (𝑥 ∈ V 𝜑) = (℩𝑥𝜑)

Proof of Theorem riotav
StepHypRef Expression
1 df-riota 7312 . 2 (𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3449 . . . 4 𝑥 ∈ V
32biantrur 531 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43iotabii 6481 . 2 (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4eqtr4i 2767 1 (𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cio 6446  crio 7311
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-in 3917  df-ss 3927  df-uni 4866  df-iota 6448  df-riota 7312
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator