MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrab2 Structured version   Unicode version

Theorem dfrab2 3624
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
Assertion
Ref Expression
dfrab2  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab2
StepHypRef Expression
1 dfrab3 3623 . 2  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
2 incom 3541 . 2  |-  ( A  i^i  { x  | 
ph } )  =  ( { x  | 
ph }  i^i  A
)
31, 2eqtri 2461 1  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   {cab 2427   {crab 2717    i^i cin 3325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-in 3333
This theorem is referenced by:  dfsup2OLD  7691  lubdm  15147  glbdm  15160  psrbagsn  17575  ismbl  21007  eulerpartgbij  26753  orvcval4  26841  dfpred3  27633  fvline2  28175
  Copyright terms: Public domain W3C validator