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

Theorem rabeq2i 2953
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 2500 . 2  |-  ( x  e.  A  <->  x  e.  { x  e.  B  |  ph } )
3 rabid 2884 . 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 2709
This theorem is referenced by:  tfis  4834  fvmptss  5813  nqereu  8806  rpnnen1lem1  10600  rpnnen1lem2  10601  rpnnen1lem3  10602  rpnnen1lem5  10604  divstgpopn  18149  ballotlem2  24746  cvmlift2lem12  25001  neibastop2lem  26389  stoweidlem24  27749  stoweidlem31  27756  stoweidlem52  27777  stoweidlem54  27779  stoweidlem57  27782  frgrawopreglem2  28434  frgrawopreg  28438  bnj1476  29218  bnj1533  29223  bnj1538  29226  bnj1523  29440
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 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-rab 2714
  Copyright terms: Public domain W3C validator