MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abbi2i Unicode version

Theorem abbi2i 2515
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
abbiri.1  |-  ( x  e.  A  <->  ph )
Assertion
Ref Expression
abbi2i  |-  A  =  { x  |  ph }
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem abbi2i
StepHypRef Expression
1 abeq2 2509 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
2 abbiri.1 . 2  |-  ( x  e.  A  <->  ph )
31, 2mpgbir 1556 1  |-  A  =  { x  |  ph }
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   {cab 2390
This theorem is referenced by:  abid2  2521  cbvralcsf  3271  cbvreucsf  3273  cbvrabcsf  3274  symdif2  3567  dfnul2  3590  dfpr2  3790  dftp2  3814  0iin  4109  epse  4525  fv3  5703  fo1st  6325  fo2nd  6326  xp2  6343  tfrlem3  6597  mapsn  7014  ixpconstg  7030  ixp0x  7049  dfom4  7560  cardnum  7931  alephiso  7935  nnzrab  10265  nn0zrab  10266  qnnen  12768  h2hcau  22435  dfch2  22862  hhcno  23360  hhcnf  23361  pjhmopidm  23639  bdayfo  25543  fobigcup  25654  dfsingles2  25674  dfrdg4  25703  tfrqfree  25704  compeq  27509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400
  Copyright terms: Public domain W3C validator