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Theorem dfrab2 3755
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 3754 . 2  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
2 incom 3661 . 2  |-  ( A  i^i  { x  | 
ph } )  =  ( { x  | 
ph }  i^i  A
)
31, 2eqtri 2458 1  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   {cab 2414   {crab 2786    i^i cin 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-in 3449
This theorem is referenced by:  dfpred3  5409  lubdm  16176  glbdm  16189  psrbagsn  18653  ismbl  22357  eulerpartgbij  29031  orvcval4  29119  fvline2  30698  nznngen  36302
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