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

Theorem rabid 2530
Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
Assertion
Ref Expression
rabid  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )

Proof of Theorem rabid
StepHypRef Expression
1 df-rab 2358 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21abeq2i 2190 1  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1434   {crab 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-rab 2358
This theorem is referenced by:  rabeq2i  2599  rabn0m  3279  repizf2lem  3943  rabxfrd  4227  onintrab2im  4270  tfis  4332
  Copyright terms: Public domain W3C validator