Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabid3 Structured version   Visualization version   GIF version

Theorem rabid3 38285
 Description: Membership in a restricted abstraction (special case). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
rabid3.1 𝐴 = {𝑥𝐵𝜑}
Assertion
Ref Expression
rabid3 (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Proof of Theorem rabid3
StepHypRef Expression
1 rabid3.1 . . 3 𝐴 = {𝑥𝐵𝜑}
21eleq2i 2680 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝐵𝜑})
3 rabid 3095 . 2 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
42, 3bitri 263 1 (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905 This theorem is referenced by:  ovolval5lem3  39544  pimdecfgtioc  39602  pimincfltioc  39603  pimdecfgtioo  39604  pimincfltioo  39605
 Copyright terms: Public domain W3C validator