MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabeq2i Structured version   Unicode version

Theorem rabeq2i 2945
Description: Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
Hypothesis
Ref Expression
rabeqi.1  |-  A  =  { x  e.  B  |  ph }
Assertion
Ref Expression
rabeq2i  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)

Proof of Theorem rabeq2i
StepHypRef Expression
1 rabeqi.1 . . 3  |-  A  =  { x  e.  B  |  ph }
21eleq2i 2499 . 2  |-  ( x  e.  A  <->  x  e.  { x  e.  B  |  ph } )
3 rabid 2876 . 2  |-  ( x  e.  { x  e.  B  |  ph }  <->  ( x  e.  B  /\  ph ) )
42, 3bitri 241 1  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701
This theorem is referenced by:  tfis  4826  fvmptss  5805  nqereu  8798  rpnnen1lem1  10592  rpnnen1lem2  10593  rpnnen1lem3  10594  rpnnen1lem5  10596  divstgpopn  18141  ballotlem2  24738  cvmlift2lem12  24993  neibastop2lem  26370  stoweidlem24  27730  stoweidlem31  27737  stoweidlem52  27758  stoweidlem54  27760  stoweidlem57  27763  frgrawopreglem2  28361  frgrawopreg  28365  bnj1476  29145  bnj1533  29150  bnj1538  29153  bnj1523  29367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-rab 2706
  Copyright terms: Public domain W3C validator