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Theorem notrab 3619
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
notrab  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem notrab
StepHypRef Expression
1 difab 3611 . 2  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  -.  ph ) }
2 difin 3579 . . 3  |-  ( A 
\  ( A  i^i  { x  |  ph }
) )  =  ( A  \  { x  |  ph } )
3 dfrab3 3618 . . . 4  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
43difeq2i 3463 . . 3  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( A 
\  ( A  i^i  { x  |  ph }
) )
5 abid2 2554 . . . 4  |-  { x  |  x  e.  A }  =  A
65difeq1i 3462 . . 3  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  ( A 
\  { x  | 
ph } )
72, 4, 63eqtr4i 2467 . 2  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( { x  |  x  e.  A }  \  {
x  |  ph }
)
8 df-rab 2715 . 2  |-  { x  e.  A  |  -.  ph }  =  { x  |  ( x  e.  A  /\  -.  ph ) }
91, 7, 83eqtr4i 2467 1  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2423   {crab 2710    \ cdif 3318    i^i cin 3320
This theorem is referenced by:  rlimrege0  12374  ordtcld1  17262  ordtcld2  17263  lhop1lem  19898  rpvmasumlem  21182  hasheuni  24476  braew  24594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rab 2715  df-v 2959  df-dif 3324  df-in 3328
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