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Theorem notrab 3404
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 3396 . 2  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  -.  ph ) }
2 difin 3364 . . 3  |-  ( A 
\  ( A  i^i  { x  |  ph }
) )  =  ( A  \  { x  |  ph } )
3 dfrab3 3403 . . . 4  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
43difeq2i 3242 . . 3  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( A 
\  ( A  i^i  { x  |  ph }
) )
5 abid2 2291 . . . 4  |-  { x  |  x  e.  A }  =  A
65difeq1i 3241 . . 3  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  ( A 
\  { x  | 
ph } )
72, 4, 63eqtr4i 2201 . 2  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( { x  |  x  e.  A }  \  {
x  |  ph }
)
8 df-rab 2457 . 2  |-  { x  e.  A  |  -.  ph }  =  { x  |  ( x  e.  A  /\  -.  ph ) }
91, 7, 83eqtr4i 2201 1  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1348    e. wcel 2141   {cab 2156   {crab 2452    \ cdif 3118    i^i cin 3120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127
This theorem is referenced by:  diffitest  6861
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