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Theorem notrab 3770
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 3762 . 2  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  -.  ph ) }
2 difin 3730 . . 3  |-  ( A 
\  ( A  i^i  { x  |  ph }
) )  =  ( A  \  { x  |  ph } )
3 dfrab3 3768 . . . 4  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
43difeq2i 3614 . . 3  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( A 
\  ( A  i^i  { x  |  ph }
) )
5 abid2 2602 . . . 4  |-  { x  |  x  e.  A }  =  A
65difeq1i 3613 . . 3  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  ( A 
\  { x  | 
ph } )
72, 4, 63eqtr4i 2501 . 2  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( { x  |  x  e.  A }  \  {
x  |  ph }
)
8 df-rab 2818 . 2  |-  { x  e.  A  |  -.  ph }  =  { x  |  ( x  e.  A  /\  -.  ph ) }
91, 7, 83eqtr4i 2501 1  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   {crab 2813    \ cdif 3468    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rab 2818  df-v 3110  df-dif 3474  df-in 3478
This theorem is referenced by:  rlimrege0  13353  ordtcld1  19459  ordtcld2  19460  lhop1lem  22144  rpvmasumlem  23395  hasheuni  27719  braew  27842
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