MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrab3 Structured version   Unicode version

Theorem dfrab3 3727
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 2765 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3720 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2544 . . 3  |-  { x  |  x  e.  A }  =  A
43ineq1i 3639 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
51, 2, 43eqtr2i 2439 1  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1407    e. wcel 1844   {cab 2389   {crab 2760    i^i cin 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-in 3423
This theorem is referenced by:  dfrab2  3728  notrab  3729  dfrab3ss  3730  dfif3  3901  dffr3  5191  dfse2  5192  tz6.26  5400  rabfi  7781  dfsup2  7938  ressmplbas2  18439  clsocv  21984  hasheuni  28545  bj-inrab3  31074  hashnzfz  36086
  Copyright terms: Public domain W3C validator