Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabab GIF version

Theorem rabab 2710
 Description: A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
rabab {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}

Proof of Theorem rabab
StepHypRef Expression
1 df-rab 2426 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 2692 . . . 4 𝑥 ∈ V
32biantrur 301 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2256 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2164 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {cab 2126  {crab 2421  Vcvv 2689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426  df-v 2691 This theorem is referenced by:  notab  3350  intmin2  3804  euen1  6703  bj-omind  13301
 Copyright terms: Public domain W3C validator